In France it is true you can go to any doctor you want, but only if you can pay for it. Many of the best doctors do not go along with the very low rates reimbursed by the Securité Sociale, and the patient is on his own.
Same for dentists.
The will pull your teeth for free, but will not do preventive dentistry or save the tooth, like root canal surgery and implants.
Yes. As in America there is gap insurance which people buy to cover the difference in reimbursement between the state insurance system and the fees of some private doctors.
I guess no-one has informed my dentist that he does not perform careful reconstruction of teeth to prevent their loss after trauma. I think I will not tell him, for fear that if he were to discover that dentists in France only extract teeth but do nothing else more sophisticated, he will stop.