From my reading of the article, WebSessions specifically doesn't do tracking. It's a replacement only for session cookies, not for tracking cookies. So with WebSessions for session maintenance, you can turn of cookies completely, if you want to avoid tracking.
It looks pretty reasonable to me (except possibly the nonce-management bit). I can't be bothered to try and pick it apart, and anyway I'm no expert.
A proposal like this will never fully replace cookies: quite apart from inertia and largely-unmaintained systems, it’s much, much more complicated for servers to implement. Diffie-Hellman, CBOR, encryption, session nonces… now compare that to just generating a big random string and using that as a cookie value and as the session table primary key. The consequence is that—even if there’s a nice backwards-compatibility path and all the browsers implement the whole lot—major frameworks like Django or Ruby on Rails will implement it, but smaller things and roll-your-own things won’t, because they’ve already got something easier that works about as well.
Just clear all cookies after the session. That's what I do, I just need to make a couple of exceptions for the few webpages I want to remember me across multiple sessions.
The extension CookieAutodelete makes it fairly painless, though you do need to remember configuring it for the webpages you actually want to stay logged in to (which may be fewer than you'd expect).
It looks pretty reasonable to me (except possibly the nonce-management bit). I can't be bothered to try and pick it apart, and anyway I'm no expert.