Are there any jurisdictions in the US with no public education for children?

Are there any jurisdictions in the United States that do not provide a free education for children who are bona-fide residents of the jurisdiction?

I recall that there was a row several decades ago in Virginia’s Prince Edward County where the County temporarily shut down the local school system rather than racially integrate it as ordered. Could such a thing legally happen today? If a jurisdiction does not provide a free public education, does the Federal Government automatically step in and provide education in a Federal school or offer tuition money for another school?

I suppose (but don’t really know) that really and truly-o uninhabited jurisdictions such as Howland Island don’t have school systems. If people actually moved there, would children be entitled to demand a free public education, either at a school in the jurisdiction itself or elsewhere?

In a nutshell, does the fact that children in the US are generally offered a free public education apply because State, Territorial, Local, etc. governments have been gracious enough to provide it, or is there an overriding Constitutional or Federal Law principle that requires it? If public education is not actually an entitlement, are there actually any jurisdictions that do not provide it?

E.g.:

Bill: “I’m thinking about moving to Baker County. Homes are cheaper there and there is a train station where I can ride into the city for work.”
Joe: “Don’t do that! One of the reasons that Baker County is so inexpensive to live in is because families don’t want to live there because Baker County does not provide public education to children. If you live there and want your kids to get an education, you have to homeschool them or send them to a private school at your expense.”

I swear, just for a second I thought this said public execution of children. Even more embarrassing, I wasn’t sure which side of the issue to come down on.

I shudder to think of the kind of community that exclusively home-schooled their children, but with the availability of the internet I trust that alternative sources are available to all but the most fervent.

Most Amish send their children to private schools but I think that even in the most dedicated communities they have a public school within a reasonable distance that they can attend. High schools would be more difficult to access in those communities because education traditionally ceases after 8th grade.

I’ve been to Lancaster County, Pennsylvania and there is definitely a mix of Amish and non-Amish to the extent that I would highly doubt there is any problem with filling high school classrooms with bona-fide Lancaster County resident schoolchildren.

Aren’t there some towns in New England that instead of operating a public high school or sending kids to a neighboring town just contract with private schools to educate students?

If the county actually funds the education and associated transportation, if necessary, then it counts as public education with respect to my OP. The jurisdiction is providing a free education to resident children. This would not apply if children can actually be denied an education because they were rejected from all private schools due to space, etc."

E.g. :

Guidance counselor: “Listen up middle schoolers. You’d better be working hard at getting into a private high school. The town will pay your tuition and transport costs if you get accepted, but if you don’t make the cut, there’s no public high school to fall back on. You’ll be flipping burgers before you know it.”

There is no federal mandate that all children be educated, but federal dollars are tied to states meeting certain educational standards.

Article 7, Section 1 of the Texas Constitution says, “A general diffusion of knowledge being essential to the preservation of the liberties and rights of the people, it shall be the duty of the Legislature of the State to establish and make suitable provision for the support and maintenance of an efficient system of public free schools.”

Every area of Texas exists within a public school district. The actual school may be quite far away, as some districts are larger than some states in other parts of the country. Nevertheless, the school is still there and available.

There are, however, several lawsuits wending their way through the system that challenge whether the Legislature of the State is actually following this provision of the Constitution, but there it is.

New York has a similar article in its Constitution. I’m pretty sure every state has a public education clause, at least since the early 20th century.

ETA:

Missouri has virtually the same languagein its Constitution. And recent court decisions have established that if a child lives in an unaccredited school district, he can attend school free of charge in an accredited district. The courts and the legislature, unfortunately, have not established just how it will be paid for, but that’s another topic.

Our problems seem to hinge on disagreement as to what “suitable provision” and “efficient” mean. Unfortunately, there is no requirement that the school system be effective.

No, our Constitution would not permit the Federal Government to do that.