That's a popular theory, but it doesn't help much because a) we want universal access (poor people just won't use it if they have to pay) b) non-professionals don't review metrics critically and don't have enough knowledge to (and the metrics are hard to interpret for experts) and c) laymen just don't have enough knowledge to make good decisions with their money, they'll just use price as a signal of quality regardless of the underlying quality.
For some ordinary, understandable things, we can get people to pay if they have the means and understanding to do so, but we need to pay for outcomes (indexed for the base sickness of the population) for more complex cases. However, paying for outcomes forces the financial risk onto the doctor, so what we really need is some mixture of "pay for performance" and "fee for service" to keep them in business, but focused on adapting their practices to evidence and modernity over the decades.
One way to reduce bureaucracy and paper work is simply "Medicare for everyone" which reduces the number of forms, data links, and creates a massive negotiating position for the payer which is needed in an inelastic market such as medicine.
To give an example, if the patients are directly on the hook and grandpa is sick, they'll just send him to the "highest quality hospital" as determined by a mixture of patient reviews ("The rooms were well lit and people smiled!") and price ("This is the most expensive and therefore best hospital in the area!"). It might even have the best metrics (the sickest and (correlated) poorest will go to the cheaper hospital and tank their metrics). They'll then pay basically anything to keep grandpa alive regardless of what the hospital asks, maybe raising money if they need to.
Hence, inelastic complex market without where transparency can actually hurt.
The other thing to note is that if people of means pay $100 cash to obtain better healthcare and jump the line, it's kind of like paying more to get gas during hurricane Sandy. You're not creating more or better resources for the people, but merely rearranging them at high cost to the benefit of the affluent. This TED talk has some interesting points about this trend:
http://www.ted.com/talks/michael_sandel_why_we_shouldn_t_tru...
Aside: I used to work in the healthcare billing industry.