There’s more than one model out there, you would know if you informed yourself about it and/or lived overseas for long enough. Mixed systems include state run and private hospitals. State run might not provide the absolute best care, so you have an option to pay for a private system that provides a better/more-personalized type of care. But that’s just one example. We ran a gamut of ideas here, from having the state provide basic care but mandate catastrophic insurance, etc. The key point, however, is cost. It’s become clear worldwide, except in the US, that the private health sector will not self-regulate. Even in countries with mixed systems, there a price controls of all types and forms. From one size fits all (Canada, UK) to lists of most common medicines and treatments (Germany) to forced licensing of generics (India).
The taxation part, sure, I don’t think the military is free either, nor are the roads and infrastructure around us. Everybody has an opinion where tax dollars should go to, that’s perfectly valid. The problem with healthcare right now in the US is that we spend way more tax dollars on it per capita than the rest of the world for similar outcomes, so while there’s an argument to be had about taxation, it’s especially important as long as the US refuses to rein in cost. Obamacare did not, so we still have that problem.