iPad Pro 1st Gen with iPadOS 16.7.10 cannot connect to FaceTime or Messages

Just for the last two days my iPad Pro 1st Gen (yes, it's updated to the latest iPadOS 16.7.10) is unable to connect to FaceTime or Messages. I noticed the issue for the first time when my send messages were not going through 2 days back.

So, I checked all the steps listed on the support pages: If you can’t turn on or sign in to iMessage or FaceTime on your iPhone - Apple Support and If you can’t turn on or sign in to iMessage or FaceTime on iPad, Mac, or Apple Vision Pro - Apple Support.

Since I turned off both the FaceTime and Messages on the iPad (as the above pages recommend), I am not able to sign in at all (I use my email login -- this AppleID works on my other devices Mac and iPhone). The error I'm getting is for FaceTime (unable to contact the FaceTime Server) and for iMessage (unable to contact the iMessage Server).

Just to reassure: my timezone is set correctly, there's no VPN configured on the iPad, the wifi works fine for everything else including accessing iCloud, AppleID etc.), and I also signed off the main AppleID from the device and re-signed back without any issue ... and I've restarted the device several times (both a soft power down and the hard restart, where you hold both the power and home key down till the iPad recycles).

Need guidance on what's amiss.