Yes I simplified. Some(? I’d hope all but probably not) new fobs do turn off (ignore the car broadcast) if they are not moved for a time. I proved this to myself with my 2020 car by putting my keys down by my car door, I could only unlock the car for a minute or two after I put it down, after that keyless entry didn’t work until I disturbed the fob to wake it up.

This is to mitigate the relay attack at home (and I’m sure other times, like if the key is in a purse), one avenue was that attackers would count on people hanging their keys by the door, so accessible to selective standing on the stoop with a relay. By turning off at rest they can’t be exploited this way.