Sure, the House has to originate all spending bills, the Senate has to sign, and then the President has to agree. or else Congress has to overcome a veto. But the President sets the broad agenda, and presents that to the public.

Or at least that's the way it's supposed to work. The Supreme Court just backed the President's right to take money Congress allocated for one thing and do something else with it. So all bets are off. If you strip the House Power of the Purse, then all they can do about the President diverting money is to impeach him.