I've waited way too long for Avistar to come and pick me up on return, but never had a problem with the car in their lot. Sometimes it's not ready for you, but that's how Avistar works.
I have since moved all my business at JFK and LGA to air park parking. It's the same type of deal, you pull in, they run you over, pick you up on the way back. At JFK they also have a full service car wash, so if you need a wash or a detail job, they do it while you are gone.
But for the actual safety of the car, never a problem at Avistar.