Might be low profit, but very stable.
Perhaps. The package portion, certainly, but other mail has fallen by quite a bit. That was one of the points brought up in the congressional hearing - volume has dropped 80% but the headcount hasn't changed.
The fact that USPS has to serve every person in the USA is the reason UPS and FedEx use them for last-mile delivery. Why run that many routes when someone else can do it for you and lose their money doing it. The post office is full of sunk costs - carriers have to run the route no matter the volume, and as volume drops the routes don't pay.
The constitution give government the power to establish a postal service, but it says nothing about letting someone else take it over. I suppose a broad reading says establish could also mean "appoint a carrier to do it, as long as that carrier has some federal oversight."
IANAL, YMMV, batteries not included, I'm not a doctor but I play one on TV, etc.