Making some progress...
- On home Wi-fi: FaceTime audio calls work fine over Wi-fi.
- On a pocket Wi-fi: FaceTime audio calls work fine over Wi-fi.
- On my friend's pocket Wi-fi: FaceTime audio calls work fine over Wi-fi.
- On work Wi-fi: iPhone flips to 4G for FaceTime audio calls, back to Wi-fi when the call is done.
I installed and control my office Wi-fi (plugged into an ethernet jack), but I don't have access to the main router/gateway in the building. I suspect ports are blocked and causing FaceTime not to work. Screen sharing has never worked using my work internet - I suspect that's a port problem as well.
Found this article on FaceTime and ports: https://support.apple.com/en-us/HT202078
Of course, can't speak for everyone, but it's leaning toward blocked ports for me and nothing hardware or iOS related.