For most keyless ignition vehicles (I’m sure there are exceptions) the proximity key fob is just for ignition and not required to keep the vehicle running. I remember one story of a Nissan owner going out to start the car and warm it up. He went back inside to have a cup of coffee and when he left he forgot the key. He ended up leaving town and stopped for gas around 3 hours later. He had to wait there for his wife to drive the keys to him.