AAQ, of course fathers can teach their kids; it is their mitzva. When they do so, let them teach how they’d like. This isn’t the question. You’re actually helping the other side: A learned father can more easily be told by school administrators, “Don’t like our system? Teach him yourself!”
My post’s intention was to explain a difference, not really to defend either side. Here are a couple of opposing arguments:
A big part of the debate going on in the secular world now is over school choice. Kids are assigned a school and they are stuck there, often despite it not providing a proper, complete education. If parents are paying taxes, and this school is their only choice in utilizing the benefits of those taxes, they claim a right to influence how the money is spent. I believe that most frum parents live in a place with a variety of schools (based on hashkafa and methodology). It becomes harder to demand that any specific school follow parental dictates. Don’t like what/how they are teaching? You have other options. What’s also true is that it’s not very likely for a yeshiva parent (who is choosing a place based on their hashkafa) to run up against a curriculum totally antithetical to their values.
The fact that it is the mitzva of the father opens the door for a counter-argument: A basic rule of shlichus is that the appointer calls the shots. The shliach has to follow instructions in order for the shlichus to work (“litikuni shadartich”). A yeshiva cannot, of course, follow every instruction given by every parent. Should the decisions be made by parental vote? By a board elected by the parents? I don’t see a problem with setting up a yeshiva that way – but I probably wouldn’t send my kids there.