No, you have to sanitize on quit. The reason cookie cleaning extensions don’t work is because they don’t clear other kinds of persistent data, eg IndexedDB etc.
TCP already partitions sites so they can’t access resources from each other and it is now the default.
Temporary containers isn’t needed anymore. You could use Firefox’s official Firefox Multi-Account Containers for isolating a domain in different contexts. For example if you had a Google Workspace account “for work” and a “Gmail account” that was your personal one, these would both share the same domains and you’d want to use MAC to separate that. (Or any other domain which you want to login multiple times to).
See arkenfox’s wonderful wiki for further details.