Who cares about power usage of the receiver in a car? It's got access to the 12v alternator/battery and as long as it turns off properly hen the car is off, radio power is going to be negligible compared to moving the vehicle, nevermind things like air conditioning, power steering, headlights, etc.

Otoh, the transmitter in your pocket might not be plugged in, so that's important.

Actually I got it backwards.

Believe it or not I don't have a smartphone and find my life is perfectly meaningful w/o it (e.g. there is nothing convenient about using cellular streaming in my car if I have to drive a few miles from home for it to work.) so I was thinking about streaming from the car audio system to headphones and not the other way around.

But generally it is the smaller, more mobile device which is going to have battery problems, which is going to be the phone if you are streaming from a phone to your car.

This is killing my in my newest stereo, the bluetooth will sleep if the phone is silent for more than 1 second, like a breathy pause during a audiobook. Then it takes a half second to kick back in. I haven't found anyone with a solution yet.

For iPhones, there is a background noise generator built into the phone (in Accessibility settings, controllable from the Control Center). On Android, a similar solution likely exists (one that continues to play even if other apps are playing audio).

You would just need to set one of those up, quiet enough not to be too annoying, but loud enough so that the stereo doesn't think its silent.

Also, there are audiobook players (for Android, at least) that skip over silent sections of the audio track.

The solution is an aux cable.

My LG OLED TV is connected via an AUX cable to a speaker, but still mutes its audio output after long-ish periods of silence. Cabled audio doesn’t help if the software sucks.

my TV now flashes "mute" every like 10 seconds in an animation... and also asks me to setup the TV every time with its cloud services...