Are you one of those people that believes most American schools are shooting ranges too?
All of the problems in the US are concentrated in subsections of the population (just as in Europe). America is a wealthy country that has a mix of south American and African problems attached to it. There is no healthcare system that is going to be able to fix this. Europe has the same problem, the difference is that share of the population was typically much smaller than the US.
I didn't say anything about the cost of healthcare on average. The US already has a public healthcare system, it doesn't work well expecting that to magically improve is not smart (again, particularly when you have evidence from other countries, even in magic Europe-land, that private healthcare can work effectively).