My guess would be the Star League doesn't limit the periphery after a much gentler Reunification War and then Amaris beats down Kerensky after he musters all the Great Houses to his side (Possibly Amaris got the CASPAR system under full control).
The first means all those small periphery nations have significant and powerful WarShip fleets and still bear a grudge over the Reunification War even if it didn't hit as hard as canon. Meanwhile, each of the Great Houses get their fleets beaten into scrap. Amaris is unable to continue his advance because although the CASPARs protect his captured worlds, those ships can't jump and invade anyone else, so he settles for the next best plan of making sure his enemies are destroyed and unable to build a second invasion fleet. Standing over the cooling wreckage of Kerensky's Orion, Stefan Amaris calls for open season on the oppressors and the Taurians, Outworlds Alliance, and Magistracy spring into action and rip bloody chunks out of the Great Houses. The battered Great Houses strip their garrisons to the bone to try to stem the tide, fruitlessly. Unbeknownst to everybody else, SLDF loyalists still on Earth uncover access codes to the CASPARs and cause the entire system to self-destruct, removing his trump card though not his surviving human-crewed ships. Amaris's dream of creating a new Star League under himself is stymied but the old one is dead and gone.
Previously, the rump SLDF and Great House fleets suffered the "Great Exodus" as hundreds of surviving units went AWOL rather than get killed in hopeless fights against CASPARs, and a smaller but significant number of Amaris units fled battle when they were badly outnumbered. These form murderously capable, excessively well-equipped pirate bands that have WarShips, fleets of DropShips and JumpShips, and multiple regiments of Battlemechs at their disposal. With many worlds defended by little more than the good ol' boy's hunting club, no small number of them decide that rather than continue on as pirates, they'll make the locals an offer they can't refuse and rip their own territories out of the Great Houses' interior where there are no defenses due to their militaries being rushed to protect their borders from the periphery. Others feel the pickings are better scooping up individual worlds on the Periphery and form their own new nations there.
When the dust settles the Rimworlders, Taurians, and Canopians hold as much territory as the Davions and Kuritas have left, and there are many dozens of small nations split among their borders, all staring enviously at each other's stuff. The time has never been better to be a mercenary.