It's not just the universal healthcare enables access to healthcare to more people. When healthcare is something being paid for by everyone, the state of other people's health matters to you too (not just your own).

Therefore, things like public smoking bans (as we have in the UK) as well as public health campaigns around alcohol consumption and healthy eating become palatable. Regulating harmful foodstuffs becomes more important. The cost of smokers' adverse health was (and still is) enormous, and reducing that burden benefits everyone.