You'd be surprised. Sure those exotic / expensive cars are targets for joy riders / certain type of thief. But the most common / mudane cars are usually the favorite targets of serial car thieves. Most don't have sophisticated alarm systems and parts from the chop shop generally won't draw much attention, can be used on a number of other cars. For the longest time - older Chevy and Ford trucks, Jeeps, Camrys, Corollas, Civics, and Accords were among the most stolen vehicles out there. These were popular and thus pretty common, so they blend easily into the crowd. If the thief stolen, say a Porsche Carrera GT, that would definitely draw people's attention. Same thief with a Corolla - basically be invisible - could drive all the way to the chop shop.
As for the key - should be fine. The immobilizer function is only if you want to start the car. If the car is equipped with an alarm, it will be pretty unhappy if you unlocked the door physically without using the remote. But you can disarm it once you opened the door to get your remote that was previously locked inside.