Well, apparently this law states otherwise:
The Presidency qualifies as "any office under the United States" is it not?
Not in the context of federal elected offices.
Article VI Clause 2 says:
This Constitution, and the Laws of the United States which shall be made in Pursuance thereof; and all Treaties made, or which shall be made, under the Authority of the United States, shall be the supreme Law of the Land; and the Judges in every State shall be bound thereby, any Thing in the Constitution or Laws of any State to the Contrary notwithstanding.The supremacy clause makes it clear that a law created by Congress is inferior to the Constitution itself and cannot add extra conditions on the qualifications for President or other Constitutional offices.
What this law refers to is this:
Article II Section 2 Clause 2:
He shall have Power, by and with the Advice and Consent of the Senate, to make Treaties, provided two thirds of the Senators present concur; and he shall nominate, and by and with the Advice and Consent of the Senate, shall appoint Ambassadors, other public Ministers and Consuls, Judges of the supreme Court, and all other Officers of the United States, whose Appointments are not herein otherwise provided for, and which shall be established by Law: but the Congress may by Law vest the Appointment of such inferior Officers, as they think proper, in the President alone, in the Courts of Law, or in the Heads of Departments.The Constitution refers to (but does not define) "inferior offices" and offices "whose Appointments are not herein otherwise provided for." Those are the offices that come under the law referenced in the OP.
-PJ
The qualifications for the office of the President are spelled out in the Constitution. I wonder if SCOTUS would rule that they are overridden by the US Code.
That bit of U.S. code does not overrule the U.S. Constitution, where the eligibility requirements for President and Vice President are clearly spelled out.