NAT has fundamental flaws that simply can't be solved, and even when it can be made to work it's still an extra layer of completely unnecessary complexity. It does work surprisingly well despite all that, but it's not something to build the entire future of the internet around.

The default config for most routers is to deny inbound connections, for both v4 and v6. I'm certain you can find routers that don't do that, but it's not common.