I'm skeptical of any supposed privacy advocate that refuses to recommend Brave. It's no Tor Browser but in my own experience and tests, it consistently wins over just about everything else outside of Tor Browser.
Anyways, it recently added a forgetful browsing feature. What I do is have that toggled on by default, turning it off on individual sites that I want to keep logged in. Ultimately, this is better for privacy AND security, since it'd limit the damage of a token stealer.
Another option might be Librewolf if you absolutely can't handle a Chromium-based browser (I also take issue with that approach but that's a different topic for a different day). Firefox but more private than default. Waterfox is also an okay option nowadays, since they're now independent from the hostile takeover that they dealt with for a while.
Overall, for my own setup, I use Brave for ~70% of my browsing, with the remaining 30% split between hardened Firefox (with BetterFox) at 20%, Librewolf at 8% and Mullvad at 2%. I only use Tor Browser once in a blue moon for sensitive subjects that could financially impact me, like medical sites, insurance research and so forth.