On using healthcare as a tool for coercion - whilst it's a debatably bad outcome, it's generally still better than healthcare locked behind money (the latter resulting a continuous, persistent denial of healthcare to some portion of the population - the former being a once off, or possibly intermittent, denial of healthcare).
Of course healthcare isn't free. The worst case, of course, is when healthcare is both paid for via taxation and then subsequent billing by non-government entities.
This sounds right, but the same issue of refusing services is just as valid (if not more valid) for private businesses. It would be a harder argument with respect to public healthcare because the first amendment is binding with respect to the government and not private companies. Said private companies and insurance companies can (and likely do) charge more because of things like vaccination.
The problem is not a matter of private vs public healthcare but of punishing violation of the first amendment and enforcing legislation that protects religious minorities from discrimination, especially on insurance companies.
The private sector is more unregulated than the public sector with regard to things like religious freedom. Corporate HR departments can get away with more coercion and subterfuge than the government, hence the current situation in the US where employers control freedom of speech and other things more thoroughly than the government ever could. It's far easier to exert power through the corporations than it is through the government
The world view is that public health care is actually a massive subsidy for corporations, who obviously benefit from a healthy (and educated) pool of workers.
That public health care is fought against in the US is more about the hideously large and inefficient health care juggernaut in the US that lobbies against the publics interest in their own favour. Forget the smokescreen that the US calls "socialism", the word has no effective meaning there other than an epithet.
Looking at this data, it seems like it pays almost 50% more than the next highest country (over 9k per capita, vs just under 6k).
In order to increase income from taxes, governments can usually most easily tax the middle class (provided middle class isn't too small of course).
And I state that free healthcare and handouts make people more dependant on the government, since governments can always provide some reason for decreasing the handouts or healthcare provided for various reasons. For example, perhaps at some point a government might decide to withheld free healthcare for civilians that refuse vaccination. Civilians at the bottom of the food chain would then feel more pressured in taking some vaccination in order to keep access to free healthcare of government handouts.
People dependant on the government will be more obedient civilians, so its likely many governments will want to move in this direction. Central Bank Digital Currencies will also give governments more control of people, so it will be the direction we'll be moving towards in the coming years.