iCloud: authentication failed

I've been trying to fix this issue for a while now; every time I try to use my iCloud account from my iPhone - for example, when I try to create an event, add something to reminders, try to send a text via iMessage - I always get rejected.

When I go in Settings, my name does appear, so I'm actually connected to my account, in fact I can use my Apple Music account, see my pics etc., but when I click on the actual iCloud setting, or on the "Payment and Shipping" button, or try to sign out, that's where the problem appears.

It keeps saying "authentication failed".

I've reached out to Apple Support, I've reset my network settings, logged in and out a thousand times, changed my password...name it, I tried it.

I'm in need to use my calendar and I can't create any event. I kinda got used to it but I'd rather find a solution.

(Yes, I can login from iCloud.com, but not from my phone.)

Help?