[comex]

> That makes it possible for a UI design where the user can press a button and say "don't surf this site anonymously" with the default being a new random session-id for all other sites.

This is already possible, just give tabs their own cookie context by default. (Browsers don't make this the default, but they all have some variant of "incognito mode" already...)

> The problem with cookies is that they disable caching of everything on the site, including favicon.ico and there is nothing the server can do about it, because the cookies are sent on all requests.

I admit that I don't know much about HTTP caching, but I don't see why the Cookie header would inhibit caching. (Edit: Isn't the purpose of the Vary header to specify which request headers affected the result, including Cookie?)