Well, yes, that goes to my point about the character and personalities of the subsequent Khans.
This goes beyond "character and personalities". Mongol taxes are not known to be light.
But, just for the sake to argument, lets say that a leader of the early Illkhanate converts to Christianity who also possesses the leadership and administrative skills of Kublai Khan.
In others words, its the best case scenario for this sort of thing. He lives a long life and leaves behind a well run and built government as well as a tradition of a Christian ruler who is tolerant of Islam, even if members of the Church of the East are obviously favored.
How does this play out; especially in his heirs, if not brilliant in their own right, are at least fairly competent.
Okay, ignoring comments on improbability and other issues because apparently we're not concerned with how outlandish this is. . .
1) You still have a Christian ruler ruling a Muslim population that is not going to be thrilled with this. "Tolerant" of them or not, he's still an infidel and a foreigner - and turning Christian is emphasizing the latter as it makes it very unlikely that the Ilkhanate will be especially receptive to Persian culture (Kublai was a Sinophile, an Iranophile is not going to be someone deliberately embracing a foreign religion from most of his subjects).
Even if he has better urban-and-fragile-agriculture policies than the OTL khans , that's going to grate.
2) You still have the Ilkhanate in conflict - probably even more so than OTL - with the Golden Horde. This is going to interfere with the kind of policies that would rebuild Iran and build Iranic acceptance of Mongol rule simply by the nature of the beast, no matter how many points we add to their Administration skill.
3) You still have the issue that the Ilkhanate's ability to fight the Mamelukes isn't nearly as impressive as vs. the already tottering Caliphate.
4) You still have the Western powers unenthusiastic about a Franco-Mongolian alliance (even if a prodigal son is better than no son at all, there's not a lot of interest even in landing in Syria).
Frankly, I think all of this - even if somehow you get a Christian khan and one who cares about the state of Iran - is going to go no better than OTL for the Ilkhanate, and if anything see more Muslims head into Anatolia.