Why would that happen with a weak Papacy? The Pope is still Patriarch over most of Europe.
But the Patriarch of Bulgaria is Patriarch of chunks of the Balkans. Or, earlier, the Patriarch of Alexandria was Patriarch of Egypt.
The fact that the Pope is a Patriarch isn't very meaningful, IMO, if he can't order anyone around. Just as autocephalous churches sprung up in the East, so you might have the same in the West. Even if the West stays united under the Patriarch of Rome, if it's a Conciliar rule rather than a Papal one, there's going to be a lot less that divides East from West.
"Better the sultan's turban. . ."
Accepting the other side is, by definition, not minimal concessions.
True, to some extent. Some large extent, even. I should certainly have said something more like "minimal possible". It would be easier, in many ways, for the East to rescue the West as a method of reunion. The OTL claim of the Papacy of supreme power was a HUGE stumbling block.
Obviously, too, the later in history, and the longer the divide lasts and the more anathemas hurled in both directions, the harder it would be to reunite the two.
Note, I never said it would be easy, because it wouldn't be. Nor even a good thing, possibly. But keeping the bulk of Christendom together for another 5-8 hundred years is probably doable.