My experience says that I don’t have the money to pay for everyone who won’t be able to pay me back. So I’ll pay for the things I need and use, and let everyone else do the same.

You quite literally pay for everyone when you pay private health insurance anyway. The math works out that either you are unfortunate enough to need care and everyone else pays for you, or you don't need extensive care and you pay for people that need it more. This is the same for privately funded or publicly funded healthcare.

The difference is that under the private healthcare system you also pay a whole bunch of salesmen, managers, investors, and executives, who can choose to delay or deny your care based on their ~~professional medical opinion~~ parasitic whim. Oh, and you also pay into all the super PACs and marketing agencies that reinforce the myth that the system is currently working for anyone.