Now for an update.
Sena customer service very quickly shipped me a package consisting of two entirely new headsets.
I pulled one out, and the first thing I checked was the version number on the label. There was none.
I thought that was odd, but I also figured that meant it was at least somehow different
from my first set. So I paired it with my phone, or tried to. I think my phone was confused by the new device with the same name, so I deleted it and was able to pair them up just fine. I put on my helmet and started the music, so far so good.
Then I paused the music, and pressed the big button to talk to someone else. Of course, I hadn't paired the damn thing with another headset, so it didn't really do much, but when I started the music again it sounded perfect!
I *had* to be sure. I pulled out the other headset and paired them together. I also paired the second one to my wife's phone. All the testing went well, with sound resuming correctly every time no matter what I did.
Then I tried getting music on my wife's new headset, and it didn't work! However, the problem was easily fixed by taking her phone back to the main screen, and off the bluetooth menu I had used to pair it up. Apparently that particular phone will only respond to the remote control from the home screen or the music app, crisis averted.
I'm quite happy with this setup now, even listening through earplugs. If and when I add to a bigger, quieter bike to the stable I can only imagine it will be even better.