This country (USA) can’t even agree to provide free universal vaccinations for children, how would they ever cover something like that for the masses? What you describe sounds like it would cost millions per person.
That sounded preposterous to me so I looked it up. I think it leans towards being false. Obamacare seems to require plans to cover them without copay, and if you don’t have health insurance, the CDC provides a free program to get vaccinations for kids.
Requiring people to buy something (insurance) and covering the cost only if their income is below a given level is not the same as providing it directly to everyone.
That’s not what this says? Either you have insurance and it must be covered or you don’t have insurance and it’s available for free through this program. Income is not a variable here.
In my hometown Columbus, Ohio, the county health department will provide free vaccinations if the parent states they are unable to pay anything. I don't know how common that is, but I don't generally consider our county or city governments to be particularly generous.