Yeah, desktop OSes give you a lot of routing options, but mobile OSes generally don't. And I'm not bringing my desktop into the shop or car with me, just to make music play without interruption.
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.)