This cookie consent functionality should be something the browser reads and gives it to you on a standard format - like the https lock and other privacy info.
I'm still not seeing the advantage. This isn't me arguing with you and telling you that you're wrong - I just genuinely don't see the difference and would like to understand better. This hypothetical service will still have the same amount of information on you either way (just stored service-side rather than in cookies), right? Unless you're claiming that the service wouldn't associate your various sessions with one another, which seems both incorrect (they certainly would, if they possibly can - as you say, via IP address, etc.) and undesirable (almost all moderately-sized-or-larger web services would feature some kind of persistent settings, at least).
Firefox does the "synchronize the authentication data across devices" thing too.
I would probably use such thing too, and not only for privacy reasons. Bonus if it deleted stored data, cache, and everything else related to the site.
Along with this we have a browser API we can call Authentication with something like Authenticate.prompt() and Authenticate.clear(). prompt() would bring up the standard browser UI for logging in and the parameters to it would dictate how the authentication should happen: username and password, 2FA options, private/public key, client certificate, etc. Registration could be handled in the same UI or have a separate API. clear() lets you have an internal logout button or mechanism on the site. The same APIs should be available via HTTP headers for non-JS usage.
As a bonus, we can then develop a mechanism for creating a session ID based on me having a private key that has an associated identity with with. So when I am prompted with a login UI instead of entering usernames and passwords I can simply choose from a drop down which identity I want to use for this site. Of course the problem of syncing private keys across devices is hard, but not any harder than what password managers currently do with my passwords.
All courts have to do is request server logs and look for this header. If it's present and the company is found to be violating people's privacy, they are obviously guilty and should be condemned and fined.
I suspect the reason Chrome doesn't do that already is that user tracking is essentially Googles business.
The downside is that I always see the cookie banners, which I mostly try to ignore but some of them block most of the page.