You are assuming that incentives to the patient are identical with or without insurance. However, a very brief analysis demonstrates that this could never be the case..
You are right, there is in fact
less incentive for patients to go routinely if there is no insurance. It is a well-known phenomena among people in the health profession. It has to do with the fact there is no natural demand for health care, it is a learned response. Worse, many people harbor an unconscious fear of it, which makes demand not only a matter of indifference, but avoidance. But such a distinction was irrelevant in my original post since either eventuality would have substantiated the point I was making.
With health insurance, for any doctor's visit, hospital stay, medical procedure, or prescription, the patient pays either a small portion of the price directly out of pocket or pays nothing out of pocket. Depending on the source of their insurance benefits, they may not even pay for the insurance premiums out of pocket, instead having them via their employer. Thus, it is impossible for the patient to ever know and accurately assess the price of healthcare and its impact on his or her finances. When the pricing does intrude on the patient's perception of healthcare consumption, it does so as the presentation that healthcare costs are so great as to be beyond the patient's capacity to pay.
The end result is the patient does not consider healthcare costs in managing their finances or their lives.
Without health insurance, the patient would be compelled to pay directly any and all healthcare costs. They would have, by virtue of the checks they write to doctors, perfect information about the impact of healthcare costs on their finances. It is impossible for a patient not to include healthcare costs in managing their finances or their lives.
So your point is? You must agree with my original assessment then. Being able to visibly see those costs would scare patients off, justifying a delay of health care. Patients, I can attest, claim to be scared off just with token co-pays. Mostly that is probably an excuse, since they often will have just finished telling you they are about to go on a trip to Tahiti. Medicine is a service they need, not want.
Having perfect knowledge of the cost would be doubly scary, even at half the current cost. Many co-pays for basic procedures are fairly small, less than someone would spend in on an impulse buy at the department store, so you have no basis to extrapolate that knowing the full, exact amount would somehow induce greater compliance, when compliance with much lower bugetable amounts is resisted.
How can we be certain of this? Easily--look at automobiles. Most car owners get their oil changed semi-regularly; they are not told to do this, but the cost savings from proper preventative care is a powerful incentive, and even when people are imperfect in maintaining their vehicles, overall they are willing to trade the inconveniences of preventative care for the cost savings of preventative care. Similarly, when one rents a vehicle, one does not worry about the oil, or the transmission fluid; there is no incentive to do so, as one does not directly bear the costs of vehicle maintenance in that circumstance..
Bad example. People are often more abusive to their bodies, although not necessarily consciously so, than they are of their cars. Also, the life-span of cars is more visibly limited, and the cost to replace it visibly high, so there is greater incentive for an oil change. That is not to say no one learns and applies a similar concern toward health care, however even then, most have no clue what the consequences of a lack of treatment would be, or even what the symptoms of something serious may be. The medical consequences of lifestyle, and therefore the true cost of medicine, will always be a nebulous, if not invisible threat to most of the population ... a threat most will not fully understand until it is too late.
A real world example: in the 60's Congress acted on the information that only about 50% of the population went to see a dentist routinely. They funded dental schools and helped push out a ton of dentists to serve the unserved. Forty years later, there still is only a 50% compliance rate. That suggests that the problem is not in the supply, but the demand. There are many well-known irrational factors in the human psyche that invalidate your hypothesis, not to mention the rational ones I have presented.
More significantly, not all serious disease is preventable. People can lower their risk, but not eliminate it. Treatment of many serious diseases is completely unaffordable already by the majority of the population. There is zero chance most could ever budget for such contingencies, ever.
Change the payment model from third party payor (insurance company) to direct payment by the patient and the incentives have to change. As incentives change, the attractiveness of preventative care will change.
Moreover, you still haven't answered the most important issue, how would prices actually fall without competition from a public system? I have never seen much price competition in any branch of medicine except in the more commodity-like cosmetic services. No one feels right about shopping around for bargain heart surgery, believe me*. The medical industry has no incentive to rebate any savings that may fall into their hands, and most doctors in the US expect to be in the upper quintile of earners. Given that the recession is pinching everyone, most doctors would seen any improvement in income as something they wholly deserve, however I suspect even in good times, many would find a way to justify keeping the extra as well.
*although if people have no choice, and if they can afford to travel overseas, sometimes take on additional medical risk and go abroad to get essential medical treatment. It's called medical tourism. That is not an ideal medical solution however, since the hidden cost is the risk inherent in travelling when ill and possible post-op complications which would necessitate expensive retreatment at home anyway, possibly then bankrupting the person.