If you decide to have a democracy or republic then you need to ensure the opportunity to participate in that democracy. The only way to effectively have an opportunity to participate is to have the mental (education) and physical (healthcare) capacity to do so. Most democracies provide for this to a reasonable extent. I think reasonable arguments can be made about what a "reasonable extent" is.

Now that I have the philosophy out of the way, I'll address "being compelled to do things by the government". The fact is, we're already compelled to do things and pay for things by the government and just about every citizen is fine with that. It's how society works / it's part of the social contract. We are compelled to pay for defense, infrastructure, fire, police, governance, and many other services. The extent to which we pay and the extent of the services provided is always up for debate but very few people argue that we shouldn't provide the aforementioned services through compelled taxation.

Now onto pragmatics. We generally put emergency services like fire, police and healthcare into the same subject. Even the U.S. does to some extent (911 emergencies, ER care for life threatening conditions, etc.). Aside from the philosophical underpinnings of why we do this, we often do this because it's the most efficient and pragmatic manner to do so. If you look up instances where police and fire were provided by private parties, paid for by individuals and not through taxation, you'll find disaster after disaster. It's not good for society to have one house policed or protected from fire while the neighbors are not. It all falls apart quite quickly.

You could argue the same with public health. Looking at the main metrics on which healthcare systems are assessed, the nations with government funded healthcare or health insurance or health insurance vouchers generally score higher in infant mortality, life expectancy, and drug and medical equipment costs while having lower administrative costs and % of GDP allocated to healthcare. So regardless of one's philosophy, if you look at the numbers, it only makes sense from a financial standpoint to go with the Beveridge, Bismark or National Insurance models vs. the out-of-pocket model. Unfortunately, the US uses about 4 different models at the same time which only increases administrative costs and reduces efficiency.

As for the attorney analogy, I don't think that really fits here (speaking as an attorney). I think my aforementioned analogies to other emergency services fits better.