On a protocol level, there is nothing in ipv6 preventing you from doing NAT. There are only less implementations of that, but it doesn't need buy-in from your ISP, as long as you control the router (and if not, you put a second router behind the first one which has your actual network).
I work for a large online service. We barely need your IP to track you. There are _so_ many other variables sites can use to track you. Even when you switch networks completely.
It's not a privacy nightmare. You could just run a proxy on your gateway and your connections would legitimately end up coming from it, but it wouldn't actually do much for your privacy.
privacy nightmare