I didn't even know until about a month ago that the USPS is one of the only constitutionally-mandated federal institutions (in school, we focused a lot on the Bill of Rights, not so much on the articles themselves).
I don't think this is quite right. The Postal Clause give congress the power to establish the USPS. It doesn't mandate that they do so. If Congress decided the USPS was no longer necessary (not a position I advocate), they would be well within their rights to pass a law abolishing it.