I'm not going to seriously argue with you because that would require me to put in effort, which I'm not going to do. Your original views on libertarianism weren't clear to me, so I guess I misinterpreted what you were saying there, though.
The burden of proof is not on me anyway. You're the one who originally claimed that the lack of public education would prevent poor people from receiving any education whatsoever. It's not on me to refute that until you actually provide proof. And no, you saying that "Well this would happen and this would happen because economics and shit EVERYONE knows is true so I'm right!" doesn't actually prove anything.
You want proof? Alright, Private schools can charge well over $50,000 a year for their services. (1)The average american household earns about $46,000 a year.(2) It is likely that a privatized system would be similar to private colleges. With tuition costs rising(3), it isn't unlikely to see the same happen to a completely privatized education system. Let me ask you, how would families be able to afford it? Student loan debt continues to pile up on America's college graduates, topping an average $29,000 per student last year. (4) How are families going to pay this on a YEARLY basis, remember that this won't be for 4 years, this will be for 12. Of course college is more expensive then K-12 but it isn't unreasonable to say that the costs for the average american family would be too much to bare. This expensive would either bankrupt families or instead their children would go without education.
Now let me go onto explain why the public option is better for all. In the United States, we allocate tax money to provide education. This tax money is collected through every american, regardless of if they have children or not. Everyone pays for education via taxation. By doing this, it makes the actual cost of education per person go down since the money is coming from more sources (the american taxpayer). According to recovery dot gov - $58,556,223,831 has been allocated to the Department of Education. The IRS collected $2.4 trillion dollars in 2009, making the percentage allocated for education 2.4 percent. (5)(These percentages rose since then). This means that on average, each american pays roughly 3% of their taxes purely for education. Every american pays about 17% of their household income on taxes. (6) Which comes to $7,820 on average per household. Which comes out to $234.60 for every american to pay purely for education tax. This of course is not an official total (just did this on a calculator) but even with a margin of error, the cost of education is significantly less if done through the public option.
http://www.privateschoolreview.com/articles/244 (1)
"Private schools charge everything from nothing at all to well over $50,000 a year for their services."
Upon further reading you find that "nothing at all" comes from students who gain admission through scholarships.
http://www.mybudget360.com/how-much-does-the-average-american-make-breaking-down-the-us-household-income-numbers/ (2)
http://money.cnn.com/2012/10/24/pf/college/public-college-tuition/index.html (3)
http://money.cnn.com/2013/12/04/pf/college/student-loan-debt/index.html (4)
http://www.answers.com/Q/What_percentage_of_federal_income_taxes_goes_to_education (5)
Of course this wouldn't be a valid source since its both old and comes from a questionable source, here is a more accurate look.
http://www2.ed.gov/about/overview/budget/index.html (5)
http://economix.blogs.nytimes.com/2009/04/08/how-much-americans-actually-pay-in-taxes/ (6)
Poll