The free market is better equipped to make that culture change happen.
A lot of people will be fucked in transition though. Even if a lot of health issues are lifestyle related, a whole bunch aren't.
30% of the current yearly healthcare expenditures in the US are a direct result of lifestyle choices. If you eliminated that, eliminated the tens of billions in yearly medicare fraud, encouraged automation/efficiency in the industry, introduced more competition, and leveraged the free market to get people to live healthier lives...you'd suddenly reduce the costs of the system so much that spending for the rest of the sick is suddenly trivial compared with the total yearly tax revenue of the US.
How do you go about doing this without fucking innocent people over in transition?