This is true and the highest level "Song of Ice and Fire" stuff would have to be butterflied away. Either Daenerys dies without getting dragons or never gets dragons or she decides that she can't even remember Westeros and just builds her own empire in Essos. The Others cut a deal with the Night's Watch involving shady shit happening at the Nightfort offscreen and don't invade.
The Wildling invasion isn't a huge threat, although it would require some support, but Stannis + the Night's Watch handle it canonically and Stannis has only kinda one Lord Paramountship.
Which is basically just meta Gary Stu issues. "Oh, we will just remove all of the big things that actually matter in the end." is
really iffy.
Jaime canonically attacks Ned in the street, seriously injuring him, before Ned gets anywhere near figuring out about Cersei's kids. And Robert doesn't really do shit about it. He pardons Ned, but Jaime doesn't get punished at all. In part that's because Catelyn did kidnap Tyrion, and later takes him to her sister, who tries to have him put to death.
Here, that's what he's imprisoned for, basically. And we know what the result was - Robert pardoning him but not moving against the Lannisters.
There is a difference between Jaime getting away with his idiocy (which really was one of the weakest parts of the series) and Cersei using authority that she doesn't have to see the sitting Hand of the King arrested on trumped up charges. If Robert wakes up, he HAS to disavow Cersei and punish her and everyone who supported her harshly or he loses the throne. Anything less loses him the North, Riverlands, and Vale; which means Dorne and the Iron Islands both declare their independence.
Everyone remembers what happened last time the royal family took unjust liberties with a Stark and the Baratheon crown is nowhere near as generally stable as the Targaryen crown was (even at the height of Aery's madness).
Cersei isn't fool enough to risk Ned's arrest so long as Ned still thinks that Joffrey is Robert's heir.
Renly and the Tyrells angle before Robert's death was actually to get Robert to set Cersei aside and marry Margaery, IIRC, so if Cersei was exposed that's more likely.
That depends on the situation, and assumes that Robert is willing to marry her. The only influence that the Tyrells had in court was Renly, Robert never liked or trusted them. Stannis instantly becomes the crown prince if Cersei's kids are removed, and he will be adamantly opposed. If the kids aren't set aside then Robert is actually better off NOT remarrying. If he does then the succession becomes a potential flash point, while if Joff is still his heir then Tywin won't move. If all else fails, Robert can boot Jamie from the Kingsguard and send him off to Tywin (along with all the Red Cloaks in Kings Landing) and basically lock Cersei up in the Maidensvault with no influence or power.
If the kids are set aside then Stannis becomes Heir once more and Robert has a war to fight against the Westerlands before he even thinks about remarrying. Robert would lead from the front, battle/war is the one thing he truly loves, and Stannis would be left in King's Landing to run the damn place. Which means that Stannis is going to gut and rebuild the court and governmental apparatus of Kings Landing and so no Tyrell friends there.
The Tyrells cutting their losses with a marriage to Renly (who would be Stannis's heir until/unless he has a son) has a great deal to recommend it in that situation.
Robert isn't dead in this scenario, so neither Stannis nor Renly can claim to be Robert's successor, because he isn't dead yet.
Robert alive and coherent makes your entire scenario farcical. There is zero way that 1) Cersei arrests Ned for treason, 2) Robert recovers and pardons Ned for that treason, 3) Cersei / the Lannisters escape punishment for that massive overreach, 4) the Stark girls remain in Kings Landing, and 5) somehow Robb becomes King in the North.
Robert, politically speaking, has to be seen to punish Cersei and whomever helped with her antics or he is done as king. No one will respect him at all and he will lose his biggest supporters. You think that the North would ever march for Robert after that? Or that the Riverlands will march to support Robert when the North opposes and Robert has allowed the Lannisters burning of the Riverlands to go unpunished?
This isn't much different from what I'm suggesting, with the exception that Jaime isn't the one to kill Ned. I am suggesting that Robert gets put out of commission for a significant length of time, and the Lannisters would use that time period to solidify their control of the capitol.
And your suggestion is basically hand wave/ authorial fiat that ignores all of the relevant political and social realities. Cersei moving against Ned with Robert conveniently in the sick bed and while rumors fly across the seven kingdoms that Lannisters poisoned Jon Arryn and Stannis has fled the capital? It's seen by everyone and their mother as an attempted Lannister coup.
Jaime killing Ned in a hotheaded duel? If Robert has Jaime killed then he makes an enemy of the Lannisters while Tywin has the closest army to the capital and the largest standing force inside Kings Landing is likely loyal to the Lannisters.
Robb will absolutely call the banners and ride south demanding Jaime's head, and will have the Riverlands at his back. The Vale won't move to stop him (indeed the reasonable assumption is that the Vale would ride in support of Robb). Dorne would love the opportunity to reduce the Lannisters so they likely through their political weight behind removing the Lannisters. The Tyrells and Stannis would both, likewise, be keen to see Lannister power reduced.
The best case for Robert is handing over Sansa, Arya, Ice, and Neds remains and getting Robb to head back North without any inconvenient questions about fealty being asked or answered. Basically, Robb is a king in all but name and everyone who matters barely even pretends the fiction that the North isn't independent.
Because hold the girls hostage and Robert loses. Dorne would take the opportunity to repudiate the Iron Throne. Greyjoy would declare his independence. The Tyrells and Renly would declare Robert a Lannister puppet and hostage, and Stannis would probably agree.
If you want to even remotely reasonably make a Robb as King in the North fic where he deals with politics more so than armies, your only realistic option is basically the following:
1) Jamie kills Ned.
2) Tywin gets to Kings Landing with an army first (probably pulling forces from the Riverlands as they are the closest) and basically takes Robert prisoner.
3) Renly camps a Stormlands army on the south bank of the Blackwater echoing the North's calls for justice and demanding Jamie's head, and calling Tywin a usurper to probably. The Tyrells start sieging the Westerlands and reinforcing Renly.
4) To remain a player in the game, Baelish gets the Vale to support the calls but publicly calls for Stannis to be recognized as Regent until such time as his brother is free and determined to be able to make his own decisions. Baelish probably slow rolls the Vale mobilization as much as possible.
5) Greyjoy declares himself King and starts attacking the Westerlands.
6) Dorne might not declare for anybody or independence but is very much anti-Lannister and all for screwing them over.
7) Seeing his position as untenable, Tywin cuts a deal with Robb and hands over the Stark girls, Ice, and Ned's remains along with probably a very hefty chunk of gold in exchange for getting his army and Robert (and whatever he can pull from King's Landing) safely through the Riverlands and back into the Westerlands via the Gold Tooth.
8) Robb declares himself King of the North and the Trident and publicly washes his hands of the whole mess with the South/Iron Throne. He secures his southern borders at key choke points and lets everyone else fight it out.
9) Stannis gets Kings Landing first. Maybe Tywin does a deal with Baelish to ensure the timing works out and Tully relations are traded on to get the Vale's forces into the Crownlands to secure them properly for Stannis.
10) Once back in the Westerlands, Tywin gives up the Iron Throne as a bad job, kills Robert, and declares himself King of the Rock.
11) Tywin offers Lannister gold (through intermediaries) to get people in the Reach, Stormlands, and Crownlands declaring for Renly/Stannis. Especially to get the Florents to declare for Stannis and condemn the Tyrells for supporting the usurper Renly.
12) Once the Baratheon civil war looks properly started, the Lannister army moves south and grabs off the chunk of land between the Mandar and the sea.
The end result is Robb as King in the North/Riverlands, Greyjoy as King in the Iron Isles, Tywin as King in the Westerlands and part of the Reach, Renly/Tyrells as King in the Stormlands and most of the Reach, and Stannis as King of the Crownlands and Vale*. Perhaps with a betrothal between Shireen and Robin. Oh, and Dorne independent again as well. Maybe Dorne marrying Arianne to Aegon/Young Griff and supporting his claim to the throne.
Perhaps Varys ensures the story of how Baelish had Lysa poison Jon Arryn and that Robin is Baelish's son spreads and is believed so as to break the Vale off from Stannis. Renly moves into the Crownlands to contest the now weakened Stannis and so Dorne moves into the Reach/Stormlands and perhaps grabs off everything south of the Mandar and maybe even sacks Highgarden. Or perhaps Renly gets a case of the dead and the Tyrells agree to support Aegon in exchange for something or other.
I do agree that the Lannisters feel a bit to weak to be proper antagonists, though. The North, Riverlands, and Iron Islands make up three of the nine Lord Paramounships, but with Dorne and the Vale sitting stuff out as the did in WOT5K that leaves only four for the Lannisters, Renly, and Stannis, with the Lannisters and Renly splitting them two each and Renly getting what are probably the more powerful of the four.
However, none of the Lord Paramountships are monoliths, and the Lannisters would have allies contesting most of the Lord Paramountships. The Iron Islands would definitely be contested, with Euron attempting a coup.
Vale could also be contested, with the Lannisters possibly betrothing Myrcella to either Sweetrobin or betrothing Myrcella to Harry the Heir and backing a faction trying to oust Lysa and Sweetrobin. Another possibility here might be for Tywin having Jaime dismissed from the Kingsguard and then betrothing him to Lysa. Notable here that Lysa isn't actually necessarily anti-Lannister, her letter to Catelyn warning her about them was an attempt to redirect blame for her own murder of her husband Jon Arryn, and Lysa will do pretty much whatever Petyr says unless she thinks it harms Sweetrobin.
No one would touch the Lannisters after the events you have laid out. Do so and you make an enemy of the North, the Riverlands, and Dorne at the least. Given the interests of Renly, Stannis, and the Tyrells you probably make an enemy of all of their factions as well.
Neither Baelish or Harry the Heir's supporters are going to be foolish enough to have their principal marry a Lannister.
And if Myrcella is still officially a Baratheon then her hand isn't going to buy armies. The best it would buy is neutrality from the Vale. They won't march to secure Joffrey's claim.
The Lannisters could absolutely be strong enough to be antagonists (although the whole point of a political story is that everyone is ally and enemy at the same time), you just need to dump the idea of them doing so with Robert still alive and actually ruling.
Tywin cutting his losses and trying to fracture the seven kingdoms so that he can be King of the Rock is entirely reasonable and has all kinds of play. Maybe he cuts a deal with the Tyrells to get the Redwyne fleet to help him conquer the Iron Isles in exchange for Tywin recognizing Mace or Renly or whomever as the King of the Stormlands and the Reach.
If that's not enough, one possibility is that Renly is murdered by the Lannisters, and Tywin reaches an understanding with the Tyrells that Joffrey's betrothal to Sansa will be broken, and that he'll wed Margaery instead. Alternatively, they make the same arrangement with another powerful Reach family in order to contest the region. Maybe a Hightower or a Redwyne. If Renly dies then the Lannisters try to have Tommen given the Stormlands, or failing that have Joffrey given Dragonstone. If Renly doesn't die, then they try to court some allies in the Stormlands and have Robert shore up Joffrey's position there.
Or the Tyrells cut a deal with Dorne, the North, the Riverlands, and the Iron Isles. The Tyrells will let a Dornish army through to take their revenge on the Lannisters, with all the loot that they can carry off, and the same offered to the Ironborn. The Tyrells use those forces to conquer the Westerlands for them and sweep in to claim dominion over the Westerlands. They recognize Dorne and the North/Riverlands as independent Kingdoms in exchange for the same recognition. Maybe they offer the North/Riverlands that if they were to take the Crownlands then the Reach would make a coordinated move to grab off the Stormlands.
Remember that the Florents are tied hard to Stannis, the Hightower daughter is married to Mace Tyrell, and that the Queen of Thorns is a Redwyne by birth. None of the major Reach houses are fertile ground for Lannister supportive rebellion in this generation. The Lannisters best play is to try and split the Reach between Renly and Stannis supporting factions by propping up the Florents.
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In the circumstances, claims of minor children don't matter so much. What matters is who the armies are willing to fight for and support. Baelish can largely keep the Vale neutral or he can throw it behind the North or, perhaps, Stannis. What he can't do is have it actively support the Lannisters or Cersei's children. Try and the Vale lords would rebel.
The Stormlands won't support the Lannisters against Stannis to seat Joffrey, regardless of whether or not Tommen is nominally their rightful lord.
In canon, the death of Robb is what made everything else possible. Robb lacks any credible, adult/near adult heir or even a credible regent for Bran. That neutralized both the North and Riverlands as players or threats for the Lannisters to be immediately concerned about. Put a blocking force at the Trident and ensure Frey loyalty and the Vale is a non issue as well.
Renly dead breaks the Stormlands/Reach alliance as the Stormlands will follow Stannis while the Reach won't. And the Tyrells don't have any inherent issue with a Lannister king on the Iron Throne if his Queen is a Tyrell, but they do fear a Stannis seated on that throne.
Robb alive and Renly dead would likely see the Tyrells throwing in with the North. A betrothal between Bran and Margaery, mutual recognition as independent powers, and a joint agreement to crush their mutual foes works just fine. An agreement with Dorne and suddenly its a joint invasion of the Westerlands followed by the Tyrells moving on the Stormlands at the same time the North moves on the Crownlands.
When all is said and done, you have independent Dorne in the far south. The Tyrells rule the Westerlands, Reach, and Stormlands. The Starks rule the North, Riverlands, Crownlands, Vale (if they can swing it), and Iron Isles.