The Mongol occupation of various parts of the former Kievan/Kyivan Rus’, though brutal, did help save the Rus’ statelets from a far worse fate: Catholic tyranny. This was at the height of the Crusades and the fallout of the Fourth Crusade could still be felt today.
In what bubble do you live that you come up with something like "Catholic tyranny"?
Anyway, the Mongols didn't save them from anything, they just crushed them and destroyed a large part of the former Rus.
Another thing is that it wouldn't have mattered at all if the Polish rulers had seriously gone about conquering parts of Rus prior to feudal decay. OTL mainly overthrew rulers and put down those affiliated with Poland, religion didn't bother then. For our expansion then went to the West and not to the East.
If we had conquered the East then, much of it would be Catholic today.
Besides, the first Russian prince who broke free from the leash of the Horde, Prince Halytsky, although he was Orthodox, received the crown from the Pope, strange isn't it?
the whole episode involving Sigismund III Vasa’s foolhardy attempt to become a Catholic Tsar,
He was not so much trying to be a tsar as to snatch from Moscow what belonged to the Commonwealth, and if he had tried, Hetman Zolkiewski, who was going to relieve the Polish garrison in the Kremlin in Moscow, would have had some forces that could have crushed the rebels and not almost nothing like OTL.
Besides, for what reason was he or his son to convert to Orthodoxy in a situation of complete mess in Moscow?
Moscow, no offense, but in that period it meant shit, and Orthodoxy was not necessary for anyone in Poland to be happy. Sigismund, by the way, if he seriously wanted to get the throne, he would have gotten it. Unlike his sons, he knew how to appoint both loyal and competent people to appropriate positions.