This issue is now resolved for me. I did not do anything specific to fix a problem I found, but here's what happened.
I brought the pump inside to look at it in more detail (I was planning on taking off the cover to look inside). I put it on a table, then set it on its left side, then lifted it from the bottom about 45 degrees so I could look at the types of screws it has. Set it on its side again.
After a few minutes, I stood it right back up, plugged it in, and decided to run the massage function again - longer than before this time. After about 20 seconds of no air, suddenly a burst of water and air shot out of the pipe.
I stopped it, took it outside, and it kept working the way I expected it to, so I was able to continue inflating the hot tub.