Public is when the government holds the wallet and pays vendors. Private is when they give a wallet to someone else to pay the vendors.
Actually your comment is probably more correct - adds a whole step to move the wallet. Misaligned incentives and mismanagement are probably more equal across public/private than we like to believe
It's just unfortunate that's how most administrators work. The traditional debate about public vs private usually focuses on different tradeoffs and incentives of the public - but if they are just paying market vendors it's greatly diminished.