IPv6 solves one of the reasons to use NAT. One more intractable reason is that many players (cellphone network operators, corporate networks) consider it a positive thing that individual devices are not globally accessible.

For cellphone networks it's not just a provider-side incentive, as inbound traffic will drain battery and you'd have no good way to stop it. But this only requires some sort of spam filter to be in front of the cellular link, like with a friends-based system where you keep connections to some friend's online nodes when you lock your phone, and just exchange IP+port info on both sides to just send a UDP packet to each other's IP+port from your own IP+port, punching your firewall. Theoretically you might even actively control your firewall to allow closing it off and also maintaining some permanent open entries for your friends to reach you with no indirection from their usual network(s). A provider could make money by selling (quota for) provider-side user-controlled firewall entries, and you could have your OS give out quota to apps.

It's feasible once you reach the point where it's worth the effort of implementing.

> Are routers assigning globally-routable IPs to their clients, is that already a thing?

Yes, but a sensible router will also firewall all incoming connections unless the port is explicitly opened.

The NATs will be gone, but they'd be just replaced by firewalls with "default deny incoming" policies for most users. Some users might change this, but there would be enough people using defaults that one could not rely on p2p connectivity.

(The current networks are often not set up to handle malicious incoming internet traffic, and new protocol is not going to change this)