We had a few extremely nervous minutes last week when our 7 year old grandson did not get off the school bus at our home due to a transportation department mix up. It was a tremendous relief when he was located, and we knew that he was safe.
I can imagine how upset the OP was not knowing where her child was when he had been driven in a private vehicle. It does sound like she waited a long time, and that after everyone else was back and she finally asked someone, it was still another 10 minutes.
I think that I would have rushed breathlessly to the vehicle when it arrived and said something like, "My God, is everything OK?" I'm not sure if that is polite or not, but that would have been my reaction. If she had then told me that she had picked up her own children I probably would have said, "Really? I thought that was against the rules." OP, I think that she got off easy when you only gave her the "glare of death."