I have a theory that NAT killed the open web. There was this idea at the beginning that everyone could host their own website, email, etc. But when you're behind a router, you suddenly have to be quite technical in order to set all that up on the computer in your room. So only (bored) technical people bother. It's possible this is the reason platforms came to dominate.
In an IPv6 world you could have a free program you download that gives you an interface like Squarespace except you can host it for free on the very computer you're using to make your site. Could be totally accessible to nontechnical people. Same goes (much more powerfully) for distributed protocols like Matrix. You could have encrypted messaging platforms like Signal that don't even need a handshake server; devices talk exclusively to each other. Etc. It would be a radically different internet.
But yeah, I think even if everything magically changed to v6 overnight, we'd still be stuck with NAT to a large extent because so much is tangled up in it at this point. For one, it probably makes it easier for ISPs to bill their customers. Especially the dynamic-IP side of things.
It’s always been a somewhat niche, difficult thing to run most standard servers. (Web/mail/ftp/gopher/nntp)