I finally contacted Apple support and found what appears to be an actual fix (vs. just a temporary work around, like restarting the phone).
We ended up doing a complete reset --
Settings > General > Reset > Reset All Settings
BUT BEFORE YOU DO THAT - check Notifications to see if "Show on lock screen" is turned OFF
Settings > Notifications > Music app > Show on lock screen
This was only suggested to me after doing the full reset (which is tedious and requires reconfiguring all your preferences, so try the other step first, in case it's enough to fix the problem)
This change in settings still allows the music/podcast etc to show player controls if the phone goes to sleep while you're using that app, but it won't show up if you're not using that app and won't come up every time you start the car and the bluetooth connects
I hope this helps.