Sure, but taxes are spread across the entire population. Insurance sort of kind of theoretically works like this (maybe substitute "time" for "people" that you're spreading the risk over), but the insurance industry are always going to be experts in making sure that no one is coming out ahead on that system. I'd rather have the entire population subsidize a small amount for everyone's health than randomly make a subset of the population destitute for what's often complete random bad luck.
Secondly, if you take a look at non-insured prices for health care in countries with universal health care (for the most part, the "universal" part is limited to residents and visitors still have to pay), the prices are still significantly lower than they are in the U.S. Having a single payer for insurance and strong regulations over health care enable things like price controls, which makes perfect sense in a health care context since it's never really going to work as a free market (emergency care needs to be treated immediately without the ability to "shop around", the care required isn't always evident at intake, doctors refuse to talk or consider financial aspects, etc.)
Secondly, if you take a look at non-insured prices for health care in countries with universal health care (for the most part, the "universal" part is limited to residents and visitors still have to pay), the prices are still significantly lower than they are in the U.S. Having a single payer for insurance and strong regulations over health care enable things like price controls, which makes perfect sense in a health care context since it's never really going to work as a free market (emergency care needs to be treated immediately without the ability to "shop around", the care required isn't always evident at intake, doctors refuse to talk or consider financial aspects, etc.)