That one is easy: because they are required to serve everyone.
If they were run as a normal company, they would not want to run rural routes because they're not cost-effective. Fedex does not deliver to rural Alaska. USPS does.
Government agencies are better when (a) the service being provided falls into the category of "natural monopoly", and (b) when coverage is required to be universal. Especially (b) because as long as you have to serve everyone, you should probably be accountable to everyone. That whole "by the people, for the people" thing, as opposed to "by the employees, for the shareholders" thing.
I'm not a big fan of having a universal tax for the benefit of the shareholders of some company.
You can disagree with the necessity of having a good postal system, but (a) as you mention, the Founders did not, and (b) I'd suggest you try living in someplace that does not have a well-run postal system.
I've lived in rural Alaska. It's a lot like frontier America in 1776: the USPS was often the only way to get things. I've also lived in rural Costa Rica, and the inability to get anything by mail was a sharp and unpleasant contrast.
Honestly, I see the USPS as being an excellent example of how government services should be run, although I would rather they be subsidized a bit more heavily. Service charges should be designed to prevent (or recoup the costs from) overuse; the majority of operating funds should come from taxation. Charges on services with a universal mandate are a form of hidden taxation: I'd rather be up-front about it. The idea of government agencies being run as for-profit businesses is actually a severe misunderstanding of what government is for.