Does this mean that (you think) the private sector would offer free schooling or significantly reduced pricing if nobody had to pay taxes?

Well, in this case I was just talking about taxes used to finance schools, but yes

What are the acceptance rates going to be like?

I imagine similar to private schools now days.

Who regulates that?

The market, it is a much tougher regulator than the state. Think of the bank bailouts, the market wanted to shut them down, the government wanted to bail them out.

If a family is worried about their budget and would benefit optimally from attending the nearest school, what happens if the private school denies them the ability to attend there?

You have this problem with accessibility to nearby grocery stores and everything else. I sure inconveniences will happen from time to time, but I'm sure it won't be much of a big deal.

Are you saying that nobody should pay taxes and everything should be operated privately?

Well, in this case I'm just talking about schools, but yes

How would that affect the economy and what makes you think this will benefit everyone?

Like most free market things, it will help the economy, may not benefit everybody, but will be better for everybody on average.

Are you saying that we should dismantle the government or just public education?

Well, I was just talking about government education here, I'm all for the former, but that's another discussion, the philosophical reasoning is similar, freedom works.