I found a roundabout solution, which was to buy an RF-to-coax adapter and plug the NES in through coax. Not ideal, as I’d really rather go through composite, but it suggests that either the problem on the NES was limited to the composite connections (and not something fundamental about the video processing) or that it was an issue with the TV. I suspect the latter. The TV may not have liked the signals it was getting through composite video, maybe the signal was too low-power to be detected as present. Or the TV and the NES were not syncing properly, like the TV was expecting a higher baud rate than the NES was providing, or the NES couldn’t get out all the new video information to the TV before the TV got tired of waiting and declared the connection lost. Something like that.
I’d still welcome any thoughts, for curiosity reasons, but from a practical standpoint, I’m calling this one solved!