Unfortunately, the USPS is written into law in the Constitution.
The Constitution and the Postal System | Constitution Center
President-elect Donald Trump has revived talk from his previous term of moving to privatize the United States Postal System. To be sure, such a process would be complicated and would likely require an act of Congress.constitutioncenter.org
So, unless you can change that and get the requirement(s) for a postal service out of the Constitution, it's gotta stay.
It does mean that Congress has ultimate control, which in effect does mean "run."
Article 1 section 8, clause 7 grants Congress the authority to establish post offices & post roads— “establish” doesn’t necessarily mean “run”, which should leave room to privatize under the 9A & 10A.
It does mean that Congress has ultimate control, which in effect does mean "run."