I will guess that one stipulation of federal aid to local schools is obedience to various federal laws and regulations.
The other stipulation is that teacher unions must donate 95% of their political funds to Democrat candidates.
Noting that public schools are uniquely a state power issue, consider that Justice John Marshall had officially clarified the following concerning Congress's power to lay taxes. Justice Marshall had indicated that Congress is prohibited from laying taxes in the name of state power issues.
“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 any federal aid that the feds give to the states for public schools on the stipulation of state compliance with federal laws are arguably state revenues which corrupt Congress wrongly stole from the states in the form of constitutionally indefensible federal taxes.
Furthermore, any federal laws which the feds bully the states to comply with had better be based on the limited powers which the states have expressly delegated to the feds via the Constitution.