This might be true but still sounds like the information given about public health systems is biased.

In Spain the public health system is still better than the private one and most of the long and most expensive treatments are done within the public circuit. You can wait months for "minor" issues and there is where a private insurance can help (actually I think is beneficial for the system that anyone being able to afford a complementary private insurance should pay for it)

Also, having experience in both systems I feel better when I am treated by someone with no economic interests.