Vaccines are not 100% effective. Some people are unable to take vaccinations. That's why we want to vaccinate as many people as possible - herd immunity protects the people who are unable to get vaccinated or people at that edge of efficacy.
And a child who is not vaccinated by their parent may want to get that vaccination when they're old enough to make that decision by themselves.
Why, categorically, should one be forcibly vaccinated in order to protect someone else? Sure, it's the civilized thing to do, but why is the justification to compel it?
(And yes, the "anti-vax" "movement" is disgusting - a fine example of the stupidity of crowds. Unfortunately, defending freedom often means defending scoundrels).
There's a similar argument around fire coverage. The common argument against a more libertarian approach to fire fighting is that if someone's house catches fire and is left to burn, then it will easily spread to neighbors, increasing their risk. And this the obvious physical reality, especially where things are dense.
But since those neighbors are the only concerned parties, then why (in principle) is that additional risk not theirs to bear and mitigate? It cannot cost any more than the neighbors splitting the cost of the fire protection on the unprotected building. Practically, the fire department would just pool that overhead across all customers.
Of course there is the issue of long term second-order effects - freeloading - since most likely if you didn't pay the fire department would still try to extinguish your house as quick as possible (possibly just busting it up a bit more in the process). But making coverage mandatory also ignores long term second-order effects when an organization does not have to be responsive to the people it serves (which is usually then handwaved away with the wisdom-of-crowds fallacy).
The root of the disagreement comes from different thinking about what society actually is. Is it a monolithic system we're all bound to and the only way to change it is top-down? Or is it the constructive result of the surplus that everyone is willing to bring to the table, implying that an individual should have the option to quit the game and take their ball - even if that individual then watches from the sideline or even returns to play but without their ball?
Both perspectives are needed, but every society is predisposed towards over-centralization - most people have a very hard time seeing anything besides their own perspective, so they falsely think their desires apply to everyone. Yet as demonstrated by the success of free markets, only decentralization can satisfy and incorporate minority viewpoints. When creative destruction is prevented from applying to structures within the society, then it will inevitably occur to the society as a whole.
> But making coverage mandatory also ignores long term second-order effects whereby organization becomes completely disconnected from the people it serves
I don't see how you can argue that it ignores this in the case where people advocating for mandatory fire protection coverage advocate that the agency through which the fire protection is provided be democratically accountable to the people in the covered area, which pretty much excludes anyone advocating public coverage in the modern developed world; addressing that effect is a central purpose of democratic accountability.
Democratic accountability to the served population deals directly with disconnection from the served population without appeal to either "wisdom of the crowds" (it doesn't rely on the "crowds" being correct, it relies only on them being the population served) or the "perfection" of centralized control (which seems to be a completely different issue, that might be worthy of discussing, but isn't the one you said was ignored.)
I edited right before you responded, swapping out "completely disconnected" in favor of "does not have to be responsive". The former was indeed too absolute to ever be true.
So yes democracy does create a connection between centralized government and the body politic. It's just a bit of a tenuous connection that addresses only gross problems that are understood (the inherent limitation of any centralized planning) by more than 51% of people.
And a child who is not vaccinated by their parent may want to get that vaccination when they're old enough to make that decision by themselves.