So, you can't point to any specific power granted to the feds to run schools.
I thought not.
It's a de facto power supported by its existence. If was unconstitutional it could have been challenged and declared unconstitutional years ago.
The Feds only "Run" The Air Force Academy, The Naval Academy, and West Point.
The feds dole out money to states and grants to local school districts to support their education systems. How much money a state or local district receives depends mostly on the number of students who qualify for Free and Reduced Meals (Tiltle I).
Although they SAID that they would give state and local governments funding for students who qualify for Special Education, only a small portion of federal money goes to support these federal mandates.
As with all state statues, those regarding education cannot conflict with Federal Statutes, so the "de facto power" of the Federal Government applies to ALL State/Locally administered institutions, including their schools, police forces, etc., etc.