macOS Big Sur - refuses to log into any accounts on system preferences (MacBook Air 13 Inch Mid 2013)

I've been using macOS Mojave for several years now, but recently, everything seemed to be crashing more than usual. I reset to factory settings, and instead of getting a fresh Mojave blank slate, I ended up reverting to OSX Mountain Lion.

Whereas it was fun to experience life from 2012 all over again, nothing really worked at all. I managed to upgrade to Mavericks, then to Big Sur - which is the latest MacOS my vintage beast is capable of running.

It's actually working decently; however, I cannot log in to any of my accounts on system preferences. When I try and log into my iCloud account, after putting in my username and password, the machine seems to think for a minute, then goes back to the starting point of asking for my username.

When attempting to log into google, I just get a splash screen and it hangs; and when I log into Microsoft exchange, it asks which areas I want to sync, but that's it - I can't actually click the 'next' button to move on and log in.

This means I cannot access my shared calendar, notes, email, or bookmarks which I have on my other devices.

It's also refusing to let me sign into google through system preferences, and my other accounts which I used to use are also not there.

I can use other apps (like Microsoft suite); but I'd like to be able to just log in.

Does anyone know what I may need to do to get this to work? I'd rather not go back to Mountain Lion then re-upgrade if I can avoid it. That took HOURS!

Thanks

[I am randomly logged into game centre though; I tried to delete that and see if it worked, but now I'm also locked out of that too!]