Life expectancy is a terrible measure for quality of healthcare. It ignore so many other factors. The only way it could be a suitable measure if you assumed every human in every country was the same except for the healthcare they receive. It's pretty clear they aren't - lifestyle, smoking rates, alcohol and drug abuse, accidents, suicides, genetics.
You’re far better to look at things like “survival after cancer diagnosis” where the US has some of the world’s best outcomes.
OK, so if we assume that, then is public health worse in Europe because smoking rates are higher than the US? How about higher alcohol consumption?
Using life expectancy to judge the quality of a healthcare system or quality of public health efforts is like judging car quality by how nice the body work is. It's a component, but a relatively minor one in overall quality.
