You ask very good questions I think. It goes to the root of the tacit agreements we in this society have made. One quick comment; the government does not pay for healthcare, we all do. We all pool our resources into a great pot and the various governments spend it on the public as a whole. Framed like that, your argument is strong. We have all agreed to shoulder our various portions of the burden, and ideally that should include not drinking until your liver fails, or smoking until your lungs fails...or refusing a vaccine for something that requires all of us to get, or the societal benefit does not occur.
On the other side though, assuming we all can get immunity simply by making a choice and getting a couple of needles, is there any real downside we should spend energy on talking about?
I do not think the public should pay for the irresponsibility of individuals. I would agree that if someone willingly refuses the vaccine (or doesn't do good diligence and climbs a mountain), they should reap the consequence. What does that look like in real life? They've presumably paid into the pot so are entitled to some service.