Remote starters that I have had will not let you drive the car without the key, or key fob.
However, you state that you have to shut the car off and restart it with the key after using the remote start. That has not been my experience.
After a remote start, you just turn the key to on, without shutting the car off, or in the case of a push button start you just push the button to activate the electronics in the car.
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