I see no difference between private insurance and government run except the government is constantly lambasted for overspending and medicare seems to have the lowest operating costs and payouts to healthcare providers.
which seems to be the exact metric you want.
edit: just move to a public option system and watch shit hit the fan in the market.
Contrary to popular understanding, relatively low (sometimes impressively low) overhead & even fraud rates is the norm for US government social programs. This is one case where pop-culture "common knowledge" diverges wildly from reality.
Accountable is obviously relative.
BigCo at least wants one thing you know about, money, and that makes them far less capricious and far easier to deal with than government where you don't know any of the incentives that the people who (with several layers of indirection) set the KPIs for the person you are actually dealing with.
As evil as they are I'll take Comcast over some random state org any day.
Having a public option as a backstop would probably be good but I don't think I want a public option that covers routine stuff, GP visits, a broken bone. Stuff with a well bounded upper limit like that tends to be handled well by private insurance so long as there's competition (see auto insurance for example).