The USPS isn't mandated by the Constitution, it is an exercise of discretionary powers granted the Congress by Article I, and Congress could, Constitutionally, choose not to exercise that power, just as it chooses not to exercise its power to issue letters of marquee and reprisal.

Its not going away because while members of Congress who would have liked to have killed it could sell quasi-privatization, various accounting mandates, and other hostile policies to their constituents as efficiency measures, they can't take the political backlash they'd expect from killing it outright.

It doesn't take a Constitutional Amendment to kill it, just Congress deciding that they want to badly enough to actually do it and take the political heat.