If you broaden the definition of "healthcare system," some of these things could be considered due to a bad healthcare system.

Suicide has almost no correlation with healthcare availability (or standard of living more broadly)

And it would be a gigantic stretch to say, blame motor vehicle deaths on the healthcare system.

What both of us are saying is that healthcare system is a factor in life expectancy, but you can't deduce from life expectancy how good a healthcare system is.