People will pay almost anything to stay alive and healthy, or to keep their loved ones alive and healthy; few people are willing to turn down the more expensive option even if the increased benefits are minor, and even if it means going bankrupt in the process;
Health-care costs are not very transparent, and bills are often computed after the fact; it is much easier to find out the price and quality of a washing machine, than the price and quality of a doctor. In other words, shopping around (assuming you have time and energy to do that while sick or dying) is not easy.
Rich countries outside of the US manage to spend much less money per capita on health care than the US, with similar health outcomes, and with a much lower rate of people forced into bankruptcy by medical expenses. They don't do this by pushing more of the cost of care onto the patient. Perhaps we in the US should try to learn from them.
And I think a large part of this stems from insurance companies paying for most of the cost of care (if you're insured) rather than the person who is receiving the care, which leads to a market failure because the insurance companies aren't incentivized to keep costs down like an individual would be.