I view all of these entities, (PO, mint, BPE, etc), as quasi-businesses. A true business has the ability to control costs, which for the most part these entities do not. OTOH, they do have some business attributes like customer service and some ability to increase sales accordingly. I am saying you both are right, they have aspects of both. How is that for a politician's answer?
What the government did with the Post Office, in my opinion, is a complete farce, but never the less, the Postal Service keeps its own money and is a private organization. The Mint, OTOH, is a bureau of the government, which has this strange retail operation. Ruben