The system isn't that split. In fact, it can work the other way around, in that a public doctor can send you to a private one when warranted, and the public system will then cover the cost.

In emergencies you can also walk into the ER of a private hospital and have the cost covered under the public system.

If you want to pay for a doctor to calm your hypochondria right now while small talking about something meaningless… Why not?

Also, my employer providing me with healthcare, isn't optional, it's legally mandated. If you have a job, you have the option of going to whatever private provider your employer has contracted. This is to make sure whatever sick leave you end up needing, is taken care of in a timely fashion so you can get back to work asap.

The only reason you can't just walk into a public hospital and see a doctor the way you can with a private one, is that the public sector will actually make sure you need the care then and there before spending its resources on you. It's triage, on a national scale.

The private and public sectors are integrated and inter-operable. This means the private sector hasn't become a price-gouging insurance mine-field. Instead it's more like an extension of the public system, serving as a more expensive but expedited channel, used where warranted.

I guess the rationale is that you give precedence to the people paying for the healthcare (middleclass workers) to get them back to contributing to workforce (and earning those tax euros) as soon as possible. Also the decision is done by the companies (trying to keep their employees in working condition, also a big perk when employees are comparing different employers) and not the government so you can't just decide to move the money like you just described.