Please post appropriate reference material, or links thereto, that supports your view.
The history is pretty clear on this question. The constitution, at least in the mind of its creators and ratifying powers, viewed it as creating a government of limited and enumerated powers. Enumerated powers is, according to most people's definition, a list of what the government is allowed to do.
Now, I suppose some rhetorical trick or clever sophistry can convert enumerated powers into "a list of what they cannot do." Based on your posts on this thread, I expect a naked disagreement with no supporting argument.
Brief history at History of the Bill of Rights.
Federalists believed a Bill of Rights was unnecessary because they believed that the Constitution only gave the government limited powers that were specifically listed. The government had no power to do things it was not entitled to in the Constitution. Alexander Hamilton believed that the people were not giving up any rights by accepting the Constitution. Therefore, in his view, it was not necessary to protect something which was not taken away from them with a Bill of Rights.The Federalists also believed, though, that adding a Bill of Rights could be very dangerous. If specific rights to be retained by the people were listed in the Constitution, they believed it would imply that any rights not listed were not protected and that the government would gradually encroach upon these rights.
The Anti-Federalists remained unpersuaded. They wanted guarantees for such things as freedom of speech, freedom of religion, freedom to petition the government and many others, specifically listed in the Constitution.
Constitution Article VI, paragraph 2 states
This Constitution, and the Laws of the United States which shall be made in Pursuance thereof; and all Treaties made, or which shall be made, under the Authority of the United States, shall be the supreme Law of the Land; and the Judges in every State shall be bound thereby, any Thing in the Constitution or Laws of any State to the Contrary notwithstanding.
Now, I just finished reading the entire Constitution again; and, I still didn't see anything at all that would lead me to believe that schools cannot be created by the either the state or federal government. If you think such is the case, please quote it so that we can discuss it. I am ready, willing, and able to discuss this objectively. Are you?
On the other hand, I can exhibit: Jefferson's Bill https://founders.archives.gov/documents/Jefferson/03-12-02-0095. This bill was for his state, but it didn't pass. Another form passed later.
So, show me that enumeration that excludes establishing schools.
Then, of course, I have already stated countless times in this thread that states established the schools, not the Federal government. See below, from Georgia's Constitution:
ARTICLE VIII.
EDUCATION
SECTION I.
PUBLIC EDUCATION
Paragraph I. Public education; free public education prior to college or postsecondary level; support by taxation. The provision of an adequate public education for the citizens shall be a primary obligation of the State of Georgia. Public education for the citizens prior to the college or postsecondary level shall be free and shall be provided for by taxation. The expense of other public education shall be provided for in such manner and in such amount as may be provided by law.
The Federal Constitution clearly states that each have their own Constitution and government.