So in the end it turned out that "Local Network" in Settings was off. But everything else the app does on the local network was operational (HTTP, etc). We were experimenting a while ago with techniques to try and detect if that setting was on or off (it does not appear there is any straightforward way to make that determination), and I must have left the toggle off. Sigh. But Quinn, thanks for the link above - it was very enlightening (even though we seem to have had everything defined properly).

Possibly related - when I connect to the camera in Settings, after it connects I get a yellow "warning" below the network name saying "No Internet Connection". This does not happen on my iPhone 6s running iOS 14.2