USPS is a government run organization. Anything they do has to approved by congress. Including budget, employee pay and postage rates. The USPS board of governors is appointed by the President with the consent of the senate.
However, it is not funded by the government, so technically it isn't a government agency like the Department of Defense or Department of Interior, Secret Service or FBI or BATF ect., etc.
You can ship any long gun anywhere in the US as long as it goes to an FFL. Handguns must go FFL to FFL. Regrettably there are many postal workers that don't know and understand their own policies or rules.
My wife has been a USPS employee for 35+ years and has ascended pretty high in management ranks.
I don't have any problems shipping guns USPS or getting claims for myself or my colleagues resolved, as long as they were smart enough to buy the insurance. If you do file a claim and it is paid, you must surrender the gun. You don't get the money and keep the gun. You get one or the other.
It ain't going to last forever though. Eventually the Dems are going to legislate shipping guns via USPS out, especially for us without FFLs. So pray for a Republican congress as long as you live.
