The central problem here is how to encourage the transition. You say it's a "more appetising alternative to cookies" -- how is it more attractive to Web sites, as compared to cookies?
If we could find a good answer to that, I think that it'd be possible to supplant cookies; e.g., by having browsers only send this new header when no cookies are stored for the origin.
And, I do think that there are some possibilities, because there's some information that cookies don't capture about user sessions (as you imply).
However, making something more attractive than cookies to Web sites means that it'll need to be as useful as cookies are now for them -- including for things like SSO, and yes, ad tracking.
If they don't support that, ad networks et al won't migrate to them, and will stay on cookies. Even if we find a way to force them to stop using cross-domain cookies and use a browser-generated, origin-scoped session ID instead, sites will likely find a way to pass identity around on the back end, making the privacy impact even more opaque than it is.