It's likely a very small factor, but yes.
A much, much, much large one is the societal and cultural norms they have there vs what we have here. It's not because of Universal healthcare that they consume, on average, about 250 fewer calories per person, each day. It's not because of Universal healthcare that the average
Swiss person takes 2x as many steps as an American. It's not because of Universal healthcare that their obesity rate is
less than 1/3 of ours. It's not because of Universal health care that their traffic related fatality rate is
less than half of ours (one would have to think if they have fewer traffic fatalities, they have fewer serious auto accidents which would require medical care - but also that they walk from place to place far more often than we do). It's not because of Universal health care that their gun related death rate (and thus I would have to assume overall shooting rate) is
1/4 that of our own. While maybe partially debatable, it's not because of Universal healthcare that Finland's (closest country I found on this list) overdose and drug related death rate is
less than half of ours.
When people are getting shot here 4x as much as they are there (which is for a different thread if you want to discuss that), people are overdosing over twice as much, serious traffic accidents are over twice as common, are 3x as likely to be obese, and consume nearly 100k more calories a year here......of course our healthcare system is going to be more expensive as well as have worse outcomes (like life expectancy and infant mortality).