Not enough iCloud storage, but there is
I'm posting this in case it helps someone else with the same goofy problem I had. The fix I found doesn't appear to show up anywhere else.
Every time I opened my photos, I was getting messages that uploading was paused because I did not have enough iCloud storage available. When I looked at available space I had like 23GB free in iCloud.
It turned out that I had somehow accidentally turned the camera on in my pocket and taken a very long video in HD which was nestled in between all of my photos. The file it created was too big to sync. There was plenty of space to sync the photos I took after that one, but it just stopped syncing anything at all because it couldn't sync that one thing.
If I was being cynical I would speculate that Apple may have designed it this way on purpose because the solution they proposed was spending more money to increase my iCloud storage rather than helping me to find out which file was clogging the pipeline.
In any event, if you run into this issue, check that you didn't accidentally create a gigantic video file that's too big to sync.