You'd need to add some filtering for the average 12v supply to not hear hum.
I use a car battery well past its prime and have it on charger set to run 2-3 AM.
I hear NO hum ever, unless it's on the real air signal.
Car radios on batteries are the best cheap on-air monitors possible, they are super-shielded, and it's easy to attentuate the input.
DC power isolated from the AC line is pure magic.
This is very helpful in use as an air-monitor where the local xmit RF gets into AC wiring , and the incoming rf bypassing is not perfect.
In such a case you get hum modulation on the air-monitor that's not really on the actual air signal.
This is why stations spend so much for monitors, among other things , to make sure the ONLY signal to the detector
came properly through a tuned circuit, not leaked in through the power supply.
I can't guess what happens when an On-star serail code is declared junked by the system. ???
I'll bet they've never closed that part of the loop, and unless reported stolen (maybe a problem) it should work.
Do these things have two antennas? You'd think so. Maybe just don't connect the high-freq data antenna....?
How's that sound on the AM for DX?