Mandates are probably not going to work but what would work is denying insurance claims. If someone ends up in a hospital occupying an ICU bed for an illness that could have been prevented by a vaccine widely and freely available then insurance companies (and medicare) should refuse to pay thousands of dollars for their treatment. It was their choice not to use preventative care available.
Or, for that matter, to other groups of people with an "illness that could have been prevented by a widely and freely available measure"?
A few spring to mind immediately – obesity first and foremost.
In broad terms, I agree with your approach: incentives at the personal level. Bottom-up. But in many countries (mine included, in EU), "health insurance" is really just a tax. There's no connection between your health & behaviour, and the "insurance" cost or fulfillment. So what you propose would essentially amount to privatizing health care, or at least turning it from an income tax into real insurance.