More clues. I have 3 devices and they all behave differently. The best behaving device had recently had settings re-set.
I wanted to find out why there were tracks on the phone that I didn't put there. I couldn't delete them, and they took up a lot of space. I noticed in Settings: iTunes, there is a "show all" option, which makes the cloud copies of your library appear on the phone and it instantly downloads if you play one. It actually downloads much more.
To get rid of these tracks, I turned the "show all" and "iTunes Match" off and then completely restored the device such that there were zero audio tracks on it. Then I selected and synced the playlists and podcasts I wanted.
After that, it plays much better. When I select a track or podcast, it actually plays it - most of the time. When it finishes a track, it's still a weird selection for the next track.