I know just about nothing about Apple ecosystem, but there's exactly that in Windows called App Volume and Device Preferences. The article mentions 3rd party apps only for older system versions, but I'd suspect someone wrote something like that for iOS/whatever.The first option is to have some sort of built-in mixer into the OS, that allows me to specifically route specific app outputs to different places. So, send Spotify output to the Bluetooth connected device (speaker in the shop, or car stereo, for example) and send all other audio through the phone's built-in speaker.
I figured out how to do it on my shitty bargain-bin Alcatel with ancient Android, so there likely is a solution less tedious than your profile setups. Almost willing to bet there's already something like this or better, hidden behind even less-specific name.
... and that auto manufacturers release ONE version of software for their head units, and then ignore any software updates forever. Oh wait. That's not right. I can get a software update for the head unit in my 2015 Chevy... for $632. (Literally sticking in a USB, clicking a button, and waiting for the update to complete. And it is over $600.)