What separates Libertarians from Anarchists is that they are not anti-government, they are anti-"large government"; in other words, they want only a necessary framework to maintain peace and stability, and nothing more (whatever can be privatized, in their opinion, should be). Whether this is a correct view or not is the subject of much debate, as there are both benefits and drawbacks to government-run organizations, and not all can be put in the same bucket (for example, the Department of Transportation might be more justified as a govt. organization than, say, the dept. of Education). (Don't get me wrong, I am all for public education, but the price we pay in taxes goes towards a lot of bureaucratic overhead, and public teachers get the short-end of the stick when it comes to salary).
A single payer system for public education is different than, for example, than an education system run entirely by the public sector. Typically there is quite a bit of outsourcing to the private sector anyway (in construction and maintenance of schools, etc.) But primary school education is paid for with public money in pretty much all developed countries around the world. The reason for that is:
Every developed country also has a single payer healthcare system (and now the US is joining them). The reason is that the public wants EVERYONE to have access to basic education and EVERYONE to have access to basic healthcare, regardless of income level. And they can collectively bargain better when there is a single payer. A free market actually results in higher prices. To be fair, by having so much power when negotiating with healthcare/education providers, the public needs to take care lest they stifle investment and innovation in those sectors. There is some central planning (with respect to budgets and allocation) that takes place now. But on the whole, having these services be cheaper for the public is a big advantage when needed, such as when there are more poor people.
As a result, there are certain regulations for schools that are paid for by public money. There has been a lot of action in our area (and I suspect around the US) about opening charter schools. These are schools that are funded with public money but are not subject to all the regulations that public schools are, and can take innovative approaches. For obvious reasons, they are different from private schools.
This is where you could use vouchers to send your kids. They still have to do things like have adequate transportation, serve certain groups of minorities, help underprivileged kids, and so forth.
However, I do agree with you in that I don't think there is much support in the US Constitution for a Federal Department of Education.
Well, there's quite a bit of overlap. Even "big-L Libertarians", meaning those who belong to the Libertarian Party in the US, are not infrequently actually free-market anarchists.
If we had a small government what would stop the rich from spending money to make it large? Some might argue that's exactly what happened in the US. I think the US' behavior as of late demonstrates that writing something in a "constitution" won't stop it from happening.
