I downloaded and installed iMazing, which allows me to browse the file system on the iPad. It turns out that all of the photos in question are ones that were deleted already. A while ago my wife went through it and deleted a bunch. They disappeared from the Photos app, but are still there in the file system. Apparently my wife deleted them more than 30 days ago because they are no longer in the recently deleted album, but they haven't been deleted. Unfortunately, I can only see them, but not delete them, with iMazing.
As for the backup, I don't have a particularly large hard drive, and I don't want to lose another ~30 GB every time I plug in the iPad. There isn't really anything on the iPad I would be upset to lose, so the cost of doing backups is greater than the reward. Now, if letting it backup would enable me to get rid of all these pictures, I would do it. If I make a backup, can I browse the backup? If so, I can back it up and see if those pictures are included in the backup. If not, then I can fix the problem by just resetting the iPad and restoring from the backup. If those pictures are included, then doing that won't help. However, I need to be able to see what is included in the backup first.