This morning (March 28th), I called Ring Technical Support again to see what could be done, or cancel my service if the problem could not be resolved. They ran me through their checklist again, with no improvement. During the many times that I was on hold with their Technical Support staff, I decided to see how far I could get by troubleshooting on my own.
I identified the IP address of the doorbell on my IoT LAN and looked at my router logs to see what it was doing. All the ports on the local Ring doorbell were closed, but I saw that it was making a TCP connection to ec2-54-165-203-48.compute-1.amazonaws.com [54.165.203.48] on port 9999. From my IoT LAN, I ran mtr -t -T -P 9999 ec2-54-165-203-48.compute-1.amazonaws.com and saw 0% packet loss for the first six hops (up to the HA AWS pair 99.83.116.142/99.83.116.146), but >50% packet loss to the endpoint.
So the problem was entirely within AWS!
I also noticed the average ping time to the 142/146 pair was 8.1ms, but it was 76.8ms to the endpoint. Assuming a V-factor of 2/3c, the problematic link is ~9,538 miles away from me. All of a sudden, I remembered that on February 24th, the Houthies cut some of the undersea Internet cables off the coast of Yemen, which is ~8000 miles away from me (as the crow flies).
It's pure speculation, but given the timing of the incidents and the location of the problems, I believe they are related.