If patriots would quit sitting on their hands, then it would be relatively easy to kick the federal government out of the public schools for the following simple reasons. First, not only have the states never delegated to Congress the specific powers to regulate public schools, evidenced by the Constitution's silence about public schools, but note the excerpt below.
As I have posted just short of a million times on FR (/s), Justice John Marshall had officially clarified that Congress cannot lay taxes in the name of state power issues, essentially any issue which Congress cannot justify under its Article I, Section 8-limited powers.
"Congress is not empowered to tax for those purposes which are within the exclusive province of the States." --Justice John Marshall, Gibbons v. Ogden, 1824.
So think of all the extra millions of dollars that each state could be spending to teach the Constitution in its public schools if the states stopped Congress from stealing state revenues in the form of constitunonally indefensible taxes that it is now laying on citizens. Same remedy for public healthcare.
The problem with the country is that the constitutionally guaranteed "persuit of happiness" for most patriots is evidently to complain about unconstitutionally big federal government as opposed policing Congress to keep Congress in its Section 8-limited power cage. So the Founding States arguably wasted their time writing everything that follows the happiness clause imo.
I doubt that you respect me.
You are about to incur the wrath or wintertime. You will be called names, have your character besmirched, and have your opinions denigrated.
It should be noted that wintertime's own children were only home schooled on an ad hoc basis. Also none of wintertime's own children have homeschooled their children except occasionally. Also of all his/ her defenders the nicest thing one of them could say about him/her was that wintertime's approach was "different".
1) Getting the Federal Government out of education would be a tremendous improvement for all the reasons you outline ( constitutional and educational), and all patriots should be working to encourage this.
2) The problem with the government schools is that they are socialist-model, compulsory, and single payer schools. Even if control were to be delegated to districts as small as a city block or suburban subdivision the schools would still be a socialist and single payer entitlement and eventually ( just as now) the push would be toward centralized control.
3) It is impossible to have religiously, politically, or culturally neutral education. Fundamentally, socialist-entitlement, single-payer, and compulsory government schools allow the voting mob to impose their NON-neutral religious, cultural, and political worldview on the children of the less politically powerful.