It’s fairly normal for something that is unashamedly trying to be the Avengers: Endgame of D&D. Making the stakes marginally lower doesn’t really seem any more believable, and seems object-defeating.
Anyway, if the gods don’t care, why should the players?
I feel like this is kind of coming at it backwards: the gods demonstrably
don't care in the adventure, which robs the premise of some of its internal logic. You need to dial back the scope so the fact that the gods aren't involved directly makes sense.
To put it in perspective, there's a bit where the players can literally converse with Tiamat while in Avernus. Furthermore, the dossier in the appendix establishes that Tiamat canonically created the Material Plane (Fizban's left the truth vague; this outright says she and Bahamut made the First World, and it shattered into different realities, and the existence of Sardior is a minor plot point) and says she is still suspected to have a soft spot for her creations. Yet, the situation is that Tiamat lets the players leave with the section of the Rod they need unhindered if they explain why they need it. So, in other words, you get to explicitly tell one of the setting's greater gods, who actually made a good-sized portion of the multiverse, that there's a dude who's going to destroy
literally everything, and her response is basically, "Oh chill. Good luck then, I guess."
And, look, you can make it work. The adventure gives you excuses. But the problem is that 5e is very careful to never
quite pin down some of the metaphysics of the multiverse, so you never really know what gods are actually in the Outer Planes, whether they're the gods of just one setting of the Material, or they're fundamental planar embodiments that (to paraphrase Discworld) have a lot of hats and a talent for voices. We don't even know if Tiamat is on Avernus by choice or she's a prisoner there. It's kept deliberately vague. Now, again, this is all
fine, but the adventure is asking us to accept that the whole of existence is at stake, and the most obvious question I would expect players to ask is, "Why don't we find whoever's actually running things and let them know?" And you can't give them an answer, because 5e doesn't have a canonical uber-god of good they can go hit up. Bahamut? Mystra? Pelor? Primus? The adventure hand-waves it away.
That's why my answer is to just lower the stakes. 'Very Bad Dude wants to Become a God' is still pretty epic. It's probably how a lot of campaigns that go all the way to level 20 end. It's literally how the first season of Critical Role ends - and it's even the same Very Bad Dude too! So I feel like they could have just gone with that and avoided the awkward metaphysical and cosmological questions that arise from a situation where the entire multiverse is potentially going to get blown up. That's kind of too big, even for a high-level D&D campaign. That's a situation where you'd expect all the gods to come together and summon the mightiest heroes from across space and time. You want basically every named character showing up. Xanathar and Elminster joining forces. Drizzt getting Mordenkainen to enchant his weapons or something. The Dragon of Tyr getting involved. The actual Spelljammer ferrying everyone to wherever Vecna is hiding out. The Lady of Pain appears to save the day at the last minute. All that junk. To be clear, that would
suck as an actual adventure, but it's what the premise as written demands.
My preference, therefore, is rather than try to wrangle the premise into something that makes sense with what's presented, throw out the premise altogether. And, again, it doesn't materially change anything in the adventure. You never really know how Vecna is planning to do what he's doing, and there isn't even much in the way of an explanation as to what happens if he succeeds. It's just, like, "the multiverse becomes a very bad place indeed!" Okay? I'll just narrate the complete restructuring of existence to my players, shall I? Any little vignettes I can slip in? No? Okay, great. So you dodge a lot of awkwardness if you just say that Vecna is doing something that's bad news, like trying to become a powerful god, or even that no one is quite sure what he's up to, but it's probably pretty shady on account of he's Vecna.