Thank you for providing the link and quotes!

In France, we have a mix of things. State generally covers 70% of the major public health fundamentals (long duration illness gets 100%), company benefit (provided mostly by non-profits) to complete and for intermediate stuff like orthondontics, and I think some additional private insurance for what's left uncovered.
So I think they may be exaggerating in thinking it's either one or the other. https://en.wikipedia.org/wiki/Health_care_in_France