I see, I can see the same thing happening here. I wonder what part of private navigation could be breaking the website? I have Firefox's tracking protection set to strict for normal browsing already, so I wasn't expecting a difference between private browsing and normal browsing.
My guess is that it's this issue again: https://bugzilla.mozilla.org/show_bug.cgi?id=1639542 but if that's the case I can't really blame the website here, other than that it doesn't show a fallback error message when Javascript crashes.
Based on a sibling comment, I'm guessing this website is using Firebase Auth, which uses the IndexedDB API, which Mozilla has disabled for private browsing.
