Because when I hit the logout button, the local session ID is deleted from my browser session storage (because that action would be performed by the browser and not by the website’s code under this system), so I would look like a brand new user to the site (setting aside other identifying stuff like IP address, etc.). All the session store should hold is an opaque ID for the session and it’s expiration info and it would be sent to the web server as a header (Session: djsisnxidnskxjf). The server would store all the info about you but if you don’t send that header, the server has no idea who you are.
OK - so instead of cookies storing information on who you are and sending that back to the service, you instead have your browser telling the service who you are and the service storing that information on you?
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).