That is patently false. Keyless car fobs are constantly transmitting and it is possible to shut them down for long term storage with a key combination that almost nobody ever uses or even knows about.
Not saying this is what Tesla do. For privacy reasons it would be way better to not transit until it sees the car. But battery life isn't proof of this.
The expected distance is pretty high. Officially up to 10m (like a keyless entry) but they can reach much higher distances. Because I often leave my AirTag in the house and it gets detected very often despite me not even having an iPhone :)