The “educations institution” is involved in these dictates because of the State’s governing body of legislators. The Federal govt. comes into the picture because they provide “Funding” for schools all across the Nation.
That fact is then broken down into WHO your governor is. We know there are States wherein governors are fighting two and nail both against mask mandates and forced vaccinations on educators.
The point being, NOT all State Governors are following these mandates. It’s all about WHO your governor is.

Masks: most all physicians, especially those specialists online ALL say, these masks are utterly useless. Even Dr. Fauci admitted that months ago. Weren’t you online when he said that? Cloth masks aren’t the least bit better.

Vaccines: the growing evidence, especially in Israel, is showing the largest vaccinated rate of any other nation on earth. Besides the death/severe illnesses rates, the specialists are pointing out that a very large part of the vaccinated population is showing up with Covid-19. Solution: Israeli’s are being vaccinated 2- 4 times.

The CDC supposedly counts ALL deaths and injuries from the vaccine, but they are purposely discounting these vaccine deaths/injuries to show ONLY 1% of those who have died from the virus, or have contracted heart disease, thrombosis, neurological impairments are actually being counted at VAERS.

Yes, that’s the condescending and placating answer I’ve been given over and over again by the Superintendent of Schools, except that’s not the information we’re getting from boots on the ground at the school and athletic level. It’s not optional. Not if they want to participate in indoor sports, cheer, and possibly choir, band and orchestra. The answer one parent was given from a school official when she said that unless her children had been recommended by a doctor to be tested for Covid she would decline the tests: “Well, I guess your children won’t be participating then, will they?”
This is not a convenience offered for parents. This is a barrier to equal access to educational services, resources and activities.
And still no one is answering the question: why is an educational institution involving itself in the medical affairs of students and families when they are clearly not equipped to do so? Are the schools going to start testing children for HIV/AIDs, hepatitis and tuberculosis, too, then allowing or prohibiting participation in sports and activities based on test results? Whatever happened to the concept of universal precautions? If masks are necessary for vaccinated teachers to keep from getting Covid, then ALL students are a DANGER to teachers and should be treated as such regardless of testing. OR, maybe, the teachers, who, a year ago were clamoring to get to the front of the line (ahead of the elderly and medically fragile) to get vaccinated should trust the efficacy of the vaccine and quit treating their students like carriers of the black plague.
Honestly, I wish they would just pick a narrative and stick to it. Are the vaccines effective? Then why are the children forced to wear masks? If the children are forced to wear masks and the masks are effective, why are the vaccines necessary? Pick one.
And what is the point of testing?
Oh! I know! And you know, too. The excuse they are looking for to close schools down again! Or, to deny access of education to those who do not comply.
DO NOT COMPLY!

Please see links below as they answer many of your question. It appears that testing is optional if a child has symptoms but requires the parent to sign off on testing and actually administer the test. The school just processes it for them.

https://docs.google.com/document/d/e/2PACX-1vTbdxd6EgyWMeHtfrPgci8PsjUUL6goB21oayZdykKDJ2QPGujv-o9Lj-5hyPDp9PfwDMGtthLFfNVM/pub#h.d1zsnn80n3av

https://docs.google.com/document/u/0/d/15Ov8buBh2wUYg2aF6GZbKjVbXqvbFpg5jZwiJj7eJp8/mobilebasic