For-profit is the delivery system, which should save taxpayer money. Systems where the institutions are directly administered by taxpayer funds and their attendant public employee unions also have political/financial incentives.
Exactly why would one or more middle men trying to extract profit save taxpayer money? What's your evidence?
I'm inclined to believe that.
Either way - publicly-run or private contractor, if the government is a good steward of public funds, they'll monitor well to avoid waste and remove people who are trying to unjustly enrich themselves. And if the officials are incompetent or corrupt themselves, they'll fail to recognize when those things are happen, or participate themselves. I don't see a big difference in public vs. private unless I imagine a fantasy government run only by officials who are smart and highly ethical.