Earlier quoted context omitted.
But from those 20K-45K you also pay income tax 30% (or 40%?), right? As for healthcare, at least in Barcelona, if it is not really urgent, you have to wait a month to actually gat that free visit to a doctor. Because of this, people go to paid/private doctor anyway.
For seeing your dermatologist, private is a lot more convenient and quick to get rid of that itch. If you have cancer or a serious disease, you go to the public one, even if you pay private. Not because of costs, but because there are better doctors in public than there are in private (strange but true).
The main goal of private companies is earning money, so they will try to cut costs. If a patient gets worse or the operation finds a problem or do not went as expected the patient will be quickly derived to the nearest public hospital (the insurance will be billed later for this).