How to reactivate iMessage on iPad?

Ever since the last iPad iOS update, the Messages app on my iPad is stuck in a loop and I can’t get out of it. When I open the app, a pop-up appears saying “Sign in with your Apple ID to activate iMessage.“. (Why? I’ve been using messages on this iPad for years.) When I enter my Apple ID and password and press sign in, it sits there for about two minutes and then has another pop-up saying “FaceTime Activation: unable to contact the FaceTime server. Try again“. Tapping outside of the pop-up to see the messages behind it doesn’t do anything.

I have tried all of the Apple support recommendations but many of them don’t apply:

They say go to Settings and turn off Messages (or FaceTime, for that matter). But I find no on/off switch there, only my Apple ID and a sign in, which then locks me into this waiting for verification loop for a couple of minutes before saying it couldn’t contact the server.

I have shut down and restarted several times. I can access my Apple account and Messages OK on both my MacBook and my iPhone. My Internet connection via Wi-Fi is solid. Settings says I’m on the latest software, and I’m not using any VPN service.

Why did this happen? This is the first time I’ve had an iOS issue appear after a software update. The only fix I can think of, which I’m hesitant to try, is to hold down the app icon on the home screen and select “remove app“ and then get a new version of Messages from the App Store if that’s possible.