Ok, it looks like my phone had a weird issue with its Internet connection, which I have now solved, and that lets me log in to the app and use it normally.
However, that raises another question: why couldn’t the app detect that the network connection was down and tell me that? A simple “the Ring app requires an Internet connection to work, please check your connection”, instead of (a) giving me a misleading message about certificate validation and kicking me back to the login screen, and (b) forcing me to restart my phone to stop the app screen flashing?