I think it's imminently reasonable to say that agencies shouldn't be independent of the executive, and that's the default if the legislation passed by Congress (and signed by the current executive!) doesn't specify otherwise. But Congress should, and I believe does, have the power to set things up otherwise.
(And it's pretty unclear to me how the NSF, which is ultimately a method for Congress to spend money, could relate to any Executive powers)
So it seems reasonable the Executive can police independent agencies to ensure they operate as authorized by law, but I don't see how that challenges the establishment or existence of lawful independent agencies.
NSF is one thing but say an independent agency that does things like negotiate treaties or command the military or nominate Supreme Court justices should be off the table
(But on the other hand, theoretically the Executive should be able to voluntarily authorize legislation that enables those things if they wanted to. Just saying I can see cooperation possible but a veto override probably won't work to force it on an unwilling Executive).
I agree with you about the powers you enumerated.
If you believe the NSF is independent then you are implicitly agreeing with those who claim there is a fourth 'deep state' branch of government that operates without accountability.
Separately: I cited the law that says the president can fire the head of the NSF. So I'm not sure why you'd conclude that any disagreement would be a problem for the president.