Actually there are valid arguments that one of the biggest problems with the German public health system is private insurance competing with the public ones. Essentially it's a licence for the private insurances to print money and siphon it from the system. They have the choice of only selecting the healthy and rich, so obviously they have much better risk distribution.

It also inflates health care cost, because the doctors use the private insurances to finance investments in expensive equipment. Talk to someone with private insurance, they essentially have fight off all sort of unnecessary tests even if they go in to get a flu vaccine. And no not every doctor needs an MRI in their practices. Also many other European countries have functioning health care without private insurances, which disproves the point they are necessary.