The fix for me was to visit
Apples Developer Downloads page, download and then install, actually re-install, the beta "Profile". After that reinstall, I did a
Check for Update and, voila, I no longer get
Unable to check for updates – No available updates found. Please try again later, but instead got the invitation to install Beta 2.
Interestingly, today, it is again giving me
Unable to check for updates – No available updates found. Please try again later. I see from that Downloads site that Beta 3 is not available yet. But because it is
unable to check, I suspect that I will need to do the same trick to get Beta 3. We shall soon find out!
By the way, I did report this to Apple, FB7869180. Indications are that this is known, widespread issue that is being worked on.