DirtyHal - I did almost EXACTLY the same thing. I have a 2011 LR4 HSE. Using a Bovee 1000. iPhone 7 Plus, running 13.1.2.
I’d gotten rid of Apple Music because it kept auto-playing the same song (Steve Miller Band - Abracadabra...now my least favorite song) every time I stepped in the vehicle, and I use Spotify anyway. I upgraded to iOS 13 just about the next day...so I blamed the iOS update. But, then I connected my old iPhone 4s and it worked fine.
Long and short, your description about the Apple Music reinstall, with one song downloaded, fixed it.
To fight against the auto-play, I emptied the entire Apple Music library and installed ONLY one song (Tool - Wings for Marie, Pt. 1). It starts with a pause and a soothing, rhythmic, deep base echo...almost like an indication the Bluetooth is connecting.
Thanks for the fix...cheers!