Replying in case someone else has this problem. I had a Ring Doorbell Pro for about two years. Successfully completed all the installation steps and had it working fine. But over time, the soft button cracked on both sides of the button in vertical lines. The doorbell and camera still worked but the cracks were unsightly. As the button is not user replaceable, they sent me a new (refurbished) Doorbell Pro.
I removed the old doorbell, deleted it from the app and tried to set up the new one using the same wiring. Set-up was uneventful until the part where it tries to connect to the internet via my WiFi router. After a couple of minutes trying to connect, I got an error saying the set-up failed as the Doorbell was “too far from the WiFi router.” The doorbell is about 12 feet from the router and had 4 bars signal strength. After this failure, the Doorbell Pro started flashing white (around the entire ring, not just one section). According to the support web site, flashing of the full ring in white means it is trying to do a firmware update. But it kept doing this for over an hour. I tried holding the reset button on the side for 30 seconds. This led to the light turning off, then after another 30 seconds or so the spinning light returned for a split second, then immediately switched back to flashing white light. I also physically unplugged the doorbell from its power wires for over 5 minutes and reconnected it, but it still went back to the flashing white light.
So I called Ring Support, went through all of the debugging steps again and they determined the unit was probably defective so they’re sending me another one. And, as expected, when I went back to my previous unit (with the broken button) setup completed perfectly and only took around 10 minutes.
TL#DR - In my case, the flashing white light on the Ring Doorbell Pro around the full circle for over an hour after a failed WiFi set-up was indicative of unit failure and required a replacement unit be sent.