Far and away, lifestyle is the biggest contributor. The very source I linked above claims it accounts for 74% of differences, and "access to health care accounted for much of the rest of the difference, the team found."
I'm open to contrary claims, but if you're making one at all it's a pretty loose one.
You seem to have missed my point, and keep focusing on overly narrow health intervention. Going back to my previous comment, education (through direct communication and through schools) is a channel through which lifestyle and hence health outcomes can be improved. Similarly, the UK we have a forthcoming sugar tax, as do many other countries. Cycle infrastructure. etc.
Depends how you define health care. If you include preventative medicine, education, etc. then things look a bit different.
Note sure what happens in the US, but countries that practice what you might call "socialized medicine" do a lot of that.