can be picked
>The problem is relay theft, where thief's relay the signal of your fob key inside the house to the car via a simple antenna and amplifier system. Cryptographical signing won't help.
AFAIK the attack you describe only applies to keyless entry systems (ie. you can open and start a car without having to pull your key out), which is related but not the same as an immobilizer. Transponder keys without keyless entry systems still exist on today's models, and is the default on most cars unless you opt for an upgrade.
>However, this can be fixed by adding a motion sensor that makes key fobs go into a sleep mode when they have been inactive for a minute.
That helps against someone cloning your key while you're at your desk, but it seems way easier to clone the key while the driver is walking away from the car? That way you know which car to steal and don't have to follow the victim into the building, which might be secured (eg. office building with badge system). Measuring RTT time and/or trilateration (multiple antennas inside car) should be much more reliable.