Barking up the wrong tree
The problem is not first party vs third party cookies. If third party cookies were banned, advertisers would simply make the switch to first party cookies. That would need a bit more server side plumbing but could be done easily by major sites.
I do not want to be tracked all across the web. What I want is a browser that lets me compartmentalize my cookies into sessions. Like an "Amazon" session, a "Register" session etc. Private tabs are too restrictive, because I want multiple tabs per session. Cookies are limited to one session and not shared between sessions. Then each site could track what I am doing on the site itself, but not my browsing habits elsewhere.
For example, a browser could manage one session per window. Then I could have multiple sessions/windows, and multiple tabs per session/window.
Does that exist?