The transponder in the key and the ignition do not rely on the battery in the keyless entry transmitter, energy is sent from the immobilizer antenna to the key to enable verification. You can remove the battery, the ignition immobilizer will still allow the engine to start.
The intrusion alarm will sound if you unlock and enter the vehicle without using the RKE transmitter, but this is a separate system and as you saw, the alarm is disarmed after the ignition is switched on.
Since this information is in the owners manual and everyone except George has immobilizer keys, everyone should be able to explain this.