I’m getting a ‘Preferences Error’ when attempting to set up internet accounts on my MacBook Pro

After even a security upgrade to my MacBook Pro (Early 2015 Mojave), I'm requested to re-log in to the basic account in the Internet Accounts. After I log in with my proper account name and password, I get a pop-up "Preferences Error" with the explanation that "There was an error in Internet Accounts preferences." Endlessly.

I tried booting into Safe Boot, logging out of iCloud, logging back into iCloud (which I've just done) and I've tried to set up that one account in Internet Accounts and the not-terribly-useful (by now) pop-up occurs.

What's next? Remove the basic account and set it up again? This has occurred on newer machines and newer OSes.

[Re-Titled by Moderator]