I’ve had the exact same experience. I first bought a Wired Stick Up Cam in April 2019 and installed it successfully using PoE (provided by a Netgear PoE Switch). I liked how it worked so much that I bought a second Wired Stick Up Cam. I attempted to set it up as well using PoE and the same Netgear PoE Switch (has 4 PoE ports). I could never get the 2nd Cam to set up successfully with PoE experiencing the same symptoms you described. I was sent a replacement Cam by Ring and was able to get it set up, but after about 2-3 weeks the second Cam would go offline…the first Cam still kept working all along. Ring then said that maybe my Netgear PoE Switch was the issue with the second Cam so they sent me 2 Ring PoE Adapters. I installed both of them and the first Cam again continued to work but the second Cam again experienced what you witnessed. I called Ring again yesterday and they are sending me another replacement Cam but I am concerned it will just do the same thing.
One thing of note, I just recently bought my third Wired Stick Up Cam and since I’m using it inside I decided to set it up with WiFi and USB power. It set up perfectly and is currently working just fine. When I get the next replacement Cam I was considering to set it up with WiFi / USB to make sure it sets up properly and updates the firmware properly and then trying to change it over to PoE, but I’m not sure if that will be successful.
It just seems to me that Ring has some issue with the PoE option on Cams since April 2019 or an issue if the set up does not work properly the first time with PoE.
Maybe Ring could confirm if they support a Wired Stick Up Cam being able to be switched from a WiFi / USB connection to a PoE connection on the same Cam?
Update: I received the replacement Wired Stick Up Cam. Instead of trying to set it up for PoE right away, I instead set it up with WiFi and USB for power. This worked and the Firmware was updated to current. After this, I then did a Connection Change to Ethernet and PoE and this too was successful. Maybe everything would have worked doinig Ethernet and PoE first, but from now on I’m gonna do the WiFi connection first and get the Firmware up to date.