Well.... okay. I'm going to dip my toes a little bit in the lore pool, and hope they don't get flayed.
The last playthrough I did was Redoran/Temple. The character was created specifically for that, and the Temple was actually the first faction he joined (well, I guess, after the Blades...)
From a game mechanics standpoint, about the only difference it seemed to make was that he had no real friction with the Temple. I would presume that the disposition hit he took as Nerevarine was countered by the boost he got as Patriarch. It did make some of the conversations a bit odd though, and particularly the one with the Archcanon.
From a lore standpoint, I thought it actually was a great twist. If I were writing a book of the story, I'd definitely add that to it, just for the additional drama. And I really think it makes a certain amount of sense.
Vivec is clearly conflicted regarding his god status. On the one hand, he certainly does have an enormous ego and a staggering degree of (inevitable, really) self-involvement, but on the other hand, you can see that, somewhere underneath it all, he has some amount of sincere concern for the Dunmer, and even some measureable amount of remorse, both for what he did to Nerevar and what he's done to the Dunmer. He likes to believe that he's helped the people, and undoubtedly has to at least some degree, but he has to know at some level, and at least hints that he
does know, that much of what they've suffered has been a fairly direct result of his (and Almalexia's and Sotha Sil's) self-involved bid for power. And leading up to the showdown with Dagoth Ur, and particularly after the death of Almalexia, he comes across, more than anything else, as resigned to his fate. It's ultimately just, and it seems that he understands that.
The only "mistake," in this context, that I'd say that Beth made with Morrowind (and a mistake that they rectified to some degree with Tribunal) was to not include some dialogue and such to communicate that the Temple and the gods are actually pretty much separate entities by that point. The Temple really doesn't need the gods any more in order to keep on going. Vivec's really just some multicolored guy floating around in a locked temple. He could just as easily not be there, and it wouldn't make any difference (as evidenced by the fact that you can kill him without having any permanent effect on the Morrowind main quest, other than making getting Wraithguard a bit more complicated). With Almalexia in Tribunal, that point was really driven home. The Temple continues on exactly as it did-- pretty much nobody will even believe that she's dead anyway. Go out front, and Mehra Helas is still out there, collecting donations and granting blessings. Nothing's really changed.
And though that's not communicated in Morrowind as it is in Tribunal, it's essentially the same. The Temple's business continues as it has, and if anything (though this isn't communicated either), they'd get a boost, since the only real problem with the Temple at all (other than rude Ordinators) is the legacy of the Tribunal's betrayal of Nerevar and their own self-involvement. With them either dead or again mortal (depending on how the player handles Vivec) that's gone, and what you're left with is just a Temple-- a bunch of nice buildings with pious people and a stated dedication to helping the Dunmer.
So it seems to me that it could be argued that the Patriarch actually benefits the Temple in the long run by destroying the heart and dragging Vivec back down to mortality. It's not as if he really does anything to benefit the Temple anyway, and his legacy is really a bit of a burden for them. Without him hanging around, it would be just that much easier to simply add the Nerevarine and all those events to the Temple's doctrines. I could even see a crafty Patriarch spinning the whole thing in such a way as to make it into an appealing bit of moralizing-- cast Vivec as a victim of the all-too-common lust for power who, when confronted with the example of what that lust for power could lead to, as embodied by Dagoth Ur, repented his choices and sought to rectify things by choosing to nobly sacrifice his own godhood, and ultimately himself, for the wellbeing of the Dunmer. Might even be able to pull the Ashlanders in if you played that one right....
Hope I didn't diverge too greatly from established lore here.....