Your healthcare analogy isn't good. The American health care system is dysfunctional for a variety of reasons; privatization is one aspect. There are efficiency gains to centralization under a public system, but it's not a sufficient condition.
There are plenty of countries that operate on a more private (less regulated) platform that have excellent health care opportunities, e.g. Thailand. Likewise, there are plenty of countries that operate on a publicly-funded model that have inferior healthcare, e.g. Russian Federation.
There are plenty of countries that operate on a more private (less regulated) platform that have excellent health care opportunities, e.g. Thailand. Likewise, there are plenty of countries that operate on a publicly-funded model that have inferior healthcare, e.g. Russian Federation.