These arguments have already happened. You are arguing about paternalism. These debates are common in policy classes, specifically about libertarianism. Clearly, people that believe in the freedom of choice would hate this. However to your questions:
> parents to not leaving their kids alone in a car
Infringes on the freedom of the child to live or not endure conditions beyond what a normal person should endure
> sending their kids to school
The other questions start getting more into removing the freedom of the parents to choose at the expense of the best interest of the individual being affected (in the government's point of view).
There is a lot of academic material with well defined terms about these subjects. Americans will err towards individual freedoms rather than the government directing more than will European countries (and obviously communist countries; yes you can find specific examples to contradict this statement, it's a generality, an average of all policies). But Americans are trending towards more paternalistic policies over the last fifty years (Standard disclaimer: To those who will derive an intent out of this statement, it is not supporting or not supporting it, simply an observation).
How do we measure what level of control we have/deserve on our kids?
I'm not disagreeing, just arguing that there are cases where one could argue that the government is overstepping on how we parent our kids.