I think the simple FM circuit works in an opposite way.
Interstation noise passes through the transformer and is rectified, keeping the relay activated and the alarm off. A nearby FM transmitter without modulation causes quieting in the FM radio and the relay is relaxed, setting off the alarm (or motor for your door).
Little FM transmitters can be designed to operate from a lithium coin cell or a couple of alkaline button cells as its battery, but not for very long.
Simple FM transmitter circuits such as this one aren't stable, they change frequency with temp change, battery voltage change and anything conductive that comes near them.