Isn't Canada in an international news story for euthanising people to save money?
The parent comment is about the accountability of public systems, not strictly about outcomes.
Also I would caution about using population level statistics to inform an individual choice about healthcare. India is a pretty poor country, but has a really good private medical system. If I travel to India for treatment I'm not exposing myself to the healthcare outcomes of the Indian population - despite the fact that I'm using their medical system.
The same thing applies to public vs private care for an individual within a country. You can tell me removing all private hospitals and healthcare will increase life expectancy for my population. But if I'm one of those people using the private system, I'm probably going to be worse off.
So now we have a story about lack of accountability causing the unfortunate needless death of infants, with a clear solution and clear cause and effect.
You can fix it by increasing accountability by telling the public hospital to "be more accountable" or you can require private hospital practicioners carry malpractice insurance.
Even a poorly run malpractice insurance market can produce positive results (read studies from 1975-onwards).
It's not so easy to make a public hospital accountable. You can't fire anyone, you can't sue them, and you can't reduce their budget if they don't perform.