Now, when it comes to leased facilities (owned by private parties), USPS pays property taxes.
That's an argument in favor of privatization. If it can operate in the black with no subsidies, it's not a natural monopoly. Which means that the most efficient way to get it done is through a private entity.
Still, they don't pay taxes on the properties owned by the US government. Even if they were totally privatized you'd find the successor companies making deals with states to avoid taxation.
I have no argument with the facts of what you just said, nor do I have any problem with those facts as they stand.
Now, when it comes to leased facilities (owned by private parties), USPS pays property taxes.
Yeah, indirectly. The owner of the facility pays the property taxes, which the USPS pays through their lease payments. State agencies can't tax federal agencies directly. McCulloch v. Maryland.