Once they got very far away from the house, the car should shut off. Or so I would think.
Mine will beep for a bit if I leave the car with the key. But the vehicle also works when the fob's battery is depleted (it has an RFID tag and an embedded physical key for the door). Having the car randomly shut off based on something so potentially flakey seems like a worse idea.
In the event that the actual owner of the car left their fob at their previous stop and discovers this fact 40 miles down the highway later, if the car were to stop, the driver is now stranded with a car that won't start. As it is now, as long as there is enough gas in the tank, the owner can just drive back and get it.
This is why every car company has examined it and chosen to not do it.
This feature actually saved huge inconvenience for us once. While visiting the other coast for wife's mom in the hospital, we used one of her parent's cars to drive to the airport with her brother to drive it back. We get out at the airport, get luggage, hugs, bye, head into terminal -- with the key still in her purse. Car running, doesn't notify him until too late to chase. If it stopped after 2min, he'd be stuck somewhere outside an airport 100mi away from anyone he knew. Instead, he just drove it home, got & used the other key for a few days, and we mailed back the first key when we arrived.
Things that seem reasonable at first....
It means the key has to he inserted somewhere. That makes it both safe and predictable.
(& yes, I still start my car with a key that is inserted, and mine also has a clutch & manual H-pattern 5-speed)
The way it works is reasonable. Maybe tighten up the proximity. But honestly, I miss my classic keys.