The big use of cookies for re-authentication and carrying around a session id enabled clean URLs lacking your session id as a query param, but it wouldn't be that big of a deal to lose it. You'd need to be careful with copy-pasting URLs -- but given browsers standardized on cookies they could have standardized on a sessionId param name to filter out of copy-pastes or not even display it at all similar to Chrome's proposal to not display the protocol, just a "secure" flag or not. A similar "session context established" flag could have been made. And in the age of password managers and/or having your browser remember your login, it wouldn't be that much of a loss to require logging in to everything again every time you restart your browser... maybe more of a pain with 2FA.

I don't really see it as "what we couldn't do" but "what poor usability we have overcome". I'm glad we have cookies and other forms of local storage, especially for the latter there are many other benefits. Maybe one day we'll get Web SQL.

In the meantime people can still disable cookies entirely, or at least delete them when they close the browser, both with out of the box browser settings (and I have no idea what extensions are available to do even more) and return to that less-usable (if slightly more private) experience. The crucial idea of a "user agent" is I think the biggest mindset change the web brought, it's important to keep that even if on the dev side we constantly complain about being asked to support more than one configuration of anything.