Keyless ignitions or push button starts are just that, you need the key to start the car and not necessarily to drive it. Even with a keyed car the key allows

Keyless ignitions or push button starts are just that, you need the key to start the car and not necessarily to drive it. Even with a keyed car the key allows you to turn the lock cylinder to start the car. Normally the lock cylinder also has a mechanism to hold the key in the cylinder when it's not in the off position but that mechanism has been known to break and wear out allowing the key to be removed while the car is running. Yes the car will continue to run without the key in the ignition it's all about the position of the lock cylinder. I think it would be unsafe to spontaneously power down the car when the remote is not detected there could be all kinds of valid reasons that a running car may not detect the remote. That being said I would think if the car was running and it missed pinging the remote periodically a message on the dash would be a wise idea.