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Congress has the power of the purse, that is they can authorize and appropriate funds.
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You forget one *very* important point: ...as authorized by the Constitution (by what authority?). “Foreign aide” is not an authorized expenditure of taxpayer $$.
Not that I am a proponent of blanket foreign aide, singling out foreign aide as not part of foreign policy in the constitution is like singling out the space force as not part of national defense. Aide is a tool of foreign policy, even if you or I think it would be better spent elsewhere.