You've missed the big picture. Your above "reason" doesn't matter. What matters is that treaties do not have to be made pursuant to the Constitution's government restrictions, per Article VI in the Constitution itself.
It would be manifestly contrary to the objectives of those who created the Constitution, as well as those who were responsible for the Bill of Rights - let alone alien to our entire constitutional history and tradition - to construe Article VI as permitting the United States to exercise power under an international agreement without observing constitutional prohibitions.