As a limiting case, a private system can't do it cheaper than a public one. A private system needs to skim profits off the top, which will always limit how cheap it can go
Private is not always cheaper — the NHS comes to mind as a counter-example, and is cheaper in part because the British government gets to act as a monopsony.
While competition can indeed help illuminate new solutions, competition can also come in the form of different political parties and international comparisons.
