There's no such thing as a free lunch. If you don't pay for the damage caused by doctor's mistakes, you are just offloading part of the cost of your healthcare system onto a few unlucky people. If you do pay for it but it's not paid for by the doctors, you just re-invented malpractice insurance but with a different source of funding.
Also, if health care coverage were universal, we wouldn't need big medical settlements to cover future medical costs since those costs would already be covered.
But mistakes happen. How you deal with those mistakes is an ideological question. Do patients who suffer from those mistakes deserve large settlements to the detriment of the vastly larger amount of patients that receive quality care? The US system believes that while other countries choose to lower costs at the expense of those unlucky few.
Also, the lack of universal coverage here means that a lot of medical bills go unpaid, for many reasons including bankruptcy or even people giving the emergency room a fake name when they go in. The hospitals don't just eat that cost, they amortize it into the costs paid by patients who do pay their bills. Universal coverage would, therefore, lower the cost of care per individual by eliminating the costs of trying to collect from uninsured people who can't pay.
But the biggest ideological cost of health care in the US is our insistence on including the "free" market in paying for health care. Insurers have a ~25% profit margin. A single-payer system could be very inefficient and yet still come in far below that cost.