That ideal doesn't exist, and probably can't, in the real world. In most situations, the easiest/fastest/cheapest way to 'profit' is by getting more customers' money, and spending less of it. When 'spending less' means customers get less care (the easiest way to spend less) then people will suffer.
The 'innovations' and such... well... I've no doubt some would/will happen, but how much 'innovation' can there be in insurance? We saw a lot of 'innovation' in the financial insurance markets (CDOs and such) and that wasn't a resounding success.
The bigger problem with your premise assumes that people are smart with respect to choosing between multiple private insurers each offering 'innovations' of their own. Most people wouldn't be on day 1 switching away from their current system, and it's not something that you can easily learn from or experiment with. With consumer items, I can buy them, try them, and recommend them to my friends if I like them or not. It's a lot harder to try out various insurance programs for multiple surgeries to see how each one fares in your particular situation.
Hospitals are often losing money because they're covering the cost of procedures for people who can't pay.
Long and short, hospitals and medical entities exist to help people with their most basic needs - health and life. People generally go to hospitals when they are sick or dying. Seeing 'for profit' companies making a profit from misery, illness, sickness and death isn't something that is going to sit well with most people.