Following up on this, I started using #Firefox Multi-Account Containers on desktop, and only logging in inside the colored containers, but not in the default one.
This results in a very similar UX as opening external links in private tabs by default on mobile (see the thread).
And hey, it's a very nifty tool in general, I hope this can one day arrive to the Android version too:
https://addons.mozilla.org/ru/firefox/addon/multi-account-containers/