Apple Account greyed out unless on Wi-Fi.

And here is the solution, thanks to a slightly younger colleague of mine (hi, Michael!).

A little bit embarrassing, but here we go...

What I failed to understand is this:

The "Apple Account" is your name and profile picture, at the top of the default screen, when you enter "settings" - but this "settings" is an app in itself.

You can configure through this app called "settings" restrictions on apps, including of course restrictions on the "settings" app.

This is why sometimes the hint with the "screen time" restrictions may work.

In my specific case, the usage of mobile data for the settings app was toggled off (I wonder who might have done that... must have been me, not knowing what I was doing).

I guess it makes sense that the per-application toggle of mobile data knows where the factual internet connection is even when there is a VPN "inbetween".

I assume that such settings would be stored in the Apple cloud pertaining to the phone in question, so after a re-install the settings would be wrong again.

So, to sum up, user was at fault by fiddling with the settings. Even when in this case I happened to be the user.

I saw in another thread folks suggested checking screen time settings. Well, this is not in use, thank you.

Also logging into Apple Account through mobile connection using Safari works without any problems.

The Watch can access Apple Account through mobile network.

I am reluctant to remove the phone from Apple Account and re.attach it because of all the tedious stuff which will invariable happen, and I do not see what on earth this should have anything to do with the question of whether WiFi is on or not.