You make a valid point. One of which I had not thought.
However, in terms of your logic, one could say that because our children are sent to govt schools, teachers would have the right to molest them. We do not have a choice but to send our children to school. We do have a choice to use the government school, private, or homeschool.
In relating to air travel, as you say, there is private air travel, which is then licensed by the government.
Actually, in many ways our children are molested in the public schools. How many times have you heard about bizarre searches of children and their belongings sometimes even strip searches of elementary children) that were conducted by school officials without parental consent?
In fact, there was a problem at my daughters middle school years ago with criminal implications for the perpetrator. The school had the police interrogate the children at school, including my daughter, without notifying the parents. The dangers for a child saying the wrong incriminating thing in this situation are obvious. My daughter wisely told them she would not talk not without me being present. They continued to berate and intimidate her. They never bothered to call me and I found out about it when my daughter came home. The next morning I contacted the principal and he told me had the school had the right to do it because while the kids were at school they had loco parentis custody of the child. I told him I did not care what rights he thought the school system had, if it happend again I would sue him personally and failing that he would have a bigger problem with me.
The point is, the public schools do believe they have the right to searches, and in some cases strip searches of children, and do so quite often.