There’s no way in which IPv6 is less private than IPv4
With IPv4 behind CGNAT you share an address with hundreds of other users. This won't protect you against a targeted subpoena, but tracking companies typically don't have this kind of power, so they have to resort to other fingerprinting options.
On the other hand, an IPv6 address is effectively a unique, and somewhat persistent, tracking ID, 48/56/64-bit long (ISP dependent), concatenated with some random garbage. And of course every advertiser, every tracking company and their dog know which part is random garbage; you are not going to fool anyone by rotating it with privacy extensions.