This is related only tangentially, as our doctor never asked for anything of the sort.
However, he had gone on vacation and when it was time to go to the delivery room, his replacement did not arrive at the hospital (Suizo) in time. The hospital's obstetrician on duty ended up doing the honors (together with our partera), and my wife was extremely happy with this doctor's care.
We subsequently had another couple of small but pressing issues that were of sufficient urgency to necessitate a middle-of-the night hospital visit, and ended up coming across the same doctor, who was amazing each time.
Long story short, we learned that this doctor had a private practice, and upon finding out that our previous ob/gyn had decided to stop accepting Swiss Medical, we happily switched. No regrets since.
The moral being that you can find excellent doctors on duty in the delivery room as well.
In our case, it was a private hospital - but the doctors are frequently the same ones in the public system as well.