Well, my thought has always been that if you are taking a child you know you aren't supposed to have, then you would be stupid to keep anything that was connected to that child.
There are other reasons... maybe it was someone with means and/or they wanted what he had to be what they chose for him. Maybe they had a car seat from a previous child or other child for immediate transport and could take it after that (airplane?) or maybe they were going to pass him off and didn't need it. I think there are a lot of reasons why they wouldn't take his car seat, maybe they had already bought a new car seat previously? Like a tan one! Or did the Smith's give Elizabeth the one they purchased for Gabriel? Also, if you already had a child you might have kept the car seat that child has now outgrown - so why have two?