A number of countries have universal healthcare largely acheived through private providers and insurers, e.g., through means-tested subsidies and coverage mandates.

In addition to that: Universal health care does not mandate a single provider. Where I live there are many health insurance companies to choose from, some are statuary others are private. All it means is that the statuary ones cannot deny you as long as you fulfill certain requirements ("Universal" health care is never really universal).

So I would have to spend more for the same care I have now? No thanks.

Not necessarily. For some countries mandatory public health contributions plus good private insurance, can still work out to cheaper than good private insurance alone in the US. This is because when private insurance has to compete with a universal public option, prices go down.