Indeed, if you go all-in on Apple they really do "just work", and come pretty close when you use a non-Apple BT headset with Apple gear. I can't say if this is because the standards are inadequate and Apple does some extra work, or if the BT implementations on cheap hardware are simply terrible. I kinda lean to the latter, but really either could be true.

> When I get into the car the car radio tries to connect to my phone, so it stops the wireless headphone output.

BTW if you are in California it's a crime to have an earbud in both ears or cup headset covering both ears. You're unlikely to be stopped for that specifically, but if you get stopped for erratic driving, speeding, or have an accident the cops are quite willing to load that one on too, and in the case of an accident your insurance company may not cover you. So your car may have a crappy implementation (see my comment above) or it might be attempting to do the right thing.