iCloud login: Could not Sign In Unable to validate account

My iPhone has never been able to connect to iCloud. I have a mac, though, that connects fine. And I can also connect to iCloud from a Linux computer using a browser. When I try to log the iPhone into iCloud, after accepting the password, it pops up an error box: "Could Not Sign In" "Unable to validate account". I can make purchases on the app store with the same apple ID and password, though. And when I log into iCloud using a browser, this iPhone appears as a trusted device. Also, if I sign in another device, I get a verification pop-up on the (problem) iPhone asking if I should authorize the other device. So I know the login and password are correct, and I know that iCloud already knows and trusts this iPhone. It just won't let me log the phone into iCloud.

A couple things I tried so far: (1) I changed the iCloud password (using the browser) and tried logging in again with the new password. It worked fine on the mac, but still didn't work on the iPhone. (2) I used the browser to remove the iPhone as a trusted device. Then I tried logging in -- got the two-factor authentication through the mac, entered the code on the iPhone, and the iPhone again appears as a trusted device in iCloud. But I still can't log the iPhone into iCloud.

I saw that others had posted having the same problem, and the fixes people suggested were pretty drastic like completely resetting the iPhone, which at least one reported didn't fix it anyway. How can I fix this? Please don't reply with "have you tried rebooting your phone" type responses (I have done that) -- reply only if you really know the answer.