The Tenth Amendment says this:
The powers not delegated to the United States by the Constitution, nor prohibited by it to the States, are reserved to the States respectively, or to the people.
Education is not mentioned in the constitution. Several people believe that the federal government should have no power over it whatever. Even a few Republican politicians.
As JoshBBParker mentions, it is a common tactic to use federal funding to wield power of the states. Indeed, over entities inside states such as contractors to the military, state parks, even individual school boards. The paying or withholding of federal money has been used as a cudgel to get what the federal politicians want no matter what anybody thinks.
And not just in education. It has been used, by both parties and for decades, over a huge array of subjects. Just a few include such things as marriage laws, banking regulations, drug enforcement, medical care, agriculture regulation, and a huge bunch more. If there's money or prestige in a thing, the federal government has probably found a way to leverage it. Even if the constitution explicitly says the federal government is forbidden to do any such thing. Or that it is only permitted to regulate interstate aspects of a thing.
This is also not limited to funding. If control of the direct funding is insufficient, there are numerous other regulatory and funding issues that can be manipulated. Just as an example, federal income tax laws can be manipulated to make certain fund raising activities difficult, and so hamper such things as "band camp" and various other things schools like to do. Say they sell crates of oranges to the neighborhood to raise money for athletic equipment. A single truckload of oranges could provide enough profit to the school to get one of those gymnastic mats that let the students learn that tumbling. The feds can come by and notice this, and make some pesky regulation about transporting agricultural products for charity purposes. And, hey presto, it's a lever over schools.
Some Republican politicians want to reduce the power of the federal government over education. Some think that's nonsense and want to impose their particular form of control instead of the Democratic Party's form of control. Or instead of some state government's control. A third group are happy to go along to get along, provided they get their "15 minutes of fame." As I said in a comment: "Throw money at it while I stand next to it."
These categories are quite similar for the Democratic Party politicians. It may be that the first category is a little smaller and the third category a little larger relative to the Republicans.
The usual result is a tug-of-war among the middle category. There is a fight over which set of controls will be enforced on education (or whatever aspect of society). There is usually very little said about whether the thing should be controlled by the government. Only what controls and how to enforce them.