I'll mention that my in laws have done this a few times too and unfortunately, I can rematch with them
It could be over reacting and it could not. It is clearly a safety issue. We have dealt with something similar with an otherwise great AP who was somewhat less sensitive about safety than the average uber protective american. I firmly told her that this was not to ever happen again, and it didn't. But she was a great AP otherwise. In your case, I would consider a few criteria: 1) was it made abundantly clear that she is not to drive your children without a car seat, and 2) is she great in all other ways and worth giving a stern warning and a second chance?
The more possible overreacting part is not wanting to keep the rematch AP for 2 weeks. If you take her car privileges away, are you concerned that she may put your child in danger in other ways?