I think you'll hear "He had her picked up and dropped off [at school] in the car" quite frequently in BrE.
I was picked up in his car.
I was dropped off in his car.
I was picked up and dropped off in his car.
I'm not going to pretend that it's a great grammatical construction but it's definitely used. I'm not even sure what the alternative would be when using "dropped off". We wouldn't say "He dropped me off from his car" (sounds dangerous!), or "He dropped me off with his car" (unnatural) or "He dropped me off using his car" (more unnatural). So despite the fact it doesn't make much sense "He dropped me off in his car" means "He drove me somewhere and then I got out of the car", not "He delivered me to the inside of his car".