Hmm...I'd call it more of an indirect influence. I mean, Almalexia went all bat[censored] (with all the "I'M THE GOD! I'M THE GOD!" ranting and raving) largely as a result of her own pathological desire to hold onto the power she gained through her own actions. She killed Sotha Sil, and then the Nerevarine killed her in self-defense. Azura's share of the blame for this is indirect at best.
Seeing as how the only way to defeat Dagoth Ur (thus cleaning up the mess the Tribunal wasn't able to handle) was to sever their original connection to the godhood, losing Ayem and Seht was really the best-case scenario. The Tribunal were murderers, traitors, and oath-breakers and suffered the consequences for their actions. Vivec even admitted to the Nerevarine that what they did was wrong.
While it is true that Azura could be said to be spiteful in holding a grudge against them, this wasn't an issue of pure jealousy on her part (they did break an oath and murder their king, after all). Vivec's banishment wasn't a moral act of justice, but an amoral act of revenge. There are no "good guys" here, but at least Azura was actually able to DO something about Dagoth Ur (even if it was indirect).
While he could have said what they did was wrong, it can also be intercepted as a precariousness evil. A whim to have beheld. You see, if we weren't told otherwise, we'd never now becuase history changed to fit the whim. The Dragon Broke at Red Mountain the moment Vivec and Sotha Sil and Almalexia became gods. This is why Nerevar proclaims 'Love is under your will only.' to Vivec. It is his will of right. It was made for him, just so. And he has a willing rule over it. All three of them do, actually. Just like how in Christianity, people follow the Bible becuase if they don't they go to this horrible place called Hell, and they fear it. Fear is the key. His will is fear, and in that he finds solitude. 'Love me no more, and the Lie Rock will come crashing down.' Searing winds, unbalanced magistrate, and the revival of one of Vivec's children? No thanks. Not to mention it equals the beginning of Landfall. So what do people do? They irritate themselves with this thought that it's not an evil plan, it's a good one. They can't complain, because they really don't know that the people they worship used to be mortals just like them.
But who's to say the line drawn between good and evil is really right? They're not existent in TES lore. It's all a matter of preference. So, what I'm getting at is that Vivec will say things that mean both the thing you want to hear, and the thing you mistrust ultimately...the truth from the mouth of the horse...er...more like Guar in this case. You could call it a timely provision, becuase Vivec doesn't REALLY think that he did was wrong. But he doesn't want the reincarnation of the person he wronged to think that. To shed a little light on it all, too, do remember that Almalexia didn't go mad from power, or lack thereof, but from her own thoughts. She became secluded, and no longer offered grace to the people who loved her. But she knew they'd love no matter what. Perhaps the waining of the Heart did that to her, but you can do the Tribunal MQ before Morrowind's, and thus it can be any given approximation of time from when the Heart was weakening (which I doubt that's true. I think Dagoth Ur was just getting smarter and learning how to draw out more power) to when you, the PC, free it and it's power diminishes slowly from the plane of Mundus.
But what if it was just in her nature to be cautious, overly-zealous, and just a plain old nut? Just like it was Sotha Sil's nature to be solitary and mingle with cosmic affairs, and it is Vivec's to deceive.
'To be otherwise was to betray his nature.'. Azura could have been a pawn to herself all along. I mean it really makes me wonder, if she knew that Vivec knew that she was only pulling at strings. Maybe you had her blessing. Maybe you were the Incarnate, maybe you were just lucky. It's possible, but in the end Vivec did what he did to her because it's in his nature. He hopped on her tower, pierced it, and user her tower as a base to jump from, to other aspiring heights and it probably helped him sustain himself evermore. CHIM was no longer an issue, but keeping himself in divinity in the face of himself was. Only so much power can be withheld from the vessel that needs it. Or maybe Azura just tightened the knot ever tighter, closer around Vivec's and the Temple's necks, by letting him pierce her. Maybe she took with her a part of Vivec that she happened to slip into her soul as her mouth was promptly [censored]. We know she got back into her realm by the time of the Oblivion Crisis, but do we know if Vivec ever got back onto his pedestal? Nope. He is probably just gone, though, laughing at the world as he holds it up like a small sliver of glass.