Well, it's not she's saying that the doctor loses privileges for one patient refusing an induction, it's that the doctor then won't be able to a
ttend the patient in the capacity as her doctor. And if the doctor DID try to do that, she could lose privileges. Which is pretty common, and yes, I would believe that, since it's policy a lot of places for non-OB doctors and midwives.
I went to 42 weeks, 1 day, and here's what we did as a "home induction" so that I wouldn't need to transfer care (and lose a chance at homebirth) from my midwives:
-Castor oil (this was the THIRD TIME -- it is worth trying again! I took a lot, 4 ounces(!), very early in the morning)
-amniotomy once the castor oil started causing ctx (that was what really got things going)
-baby was definitely on the far side of well-done! This was NOT a case of baby just needing to stay in longer.
-btw I never had any ctx without the castor oil. First couple times, it gave me prodromal labor but that was it.
-pm if you want more details.

It's more interventive than just waiting, of course, but it was preferable to the pitocin route for me. Sometimes a small intervention can save you a larger one down the road.
Someone upthread did make a good point, though -- your doctor usually spends very little time with you in labor. Unless you feel like she'll be the lone guardian of your birthing integrity, it might not be the end of the world to give yourself a few more days. But you have options -- whether to induce or not to, what methods to use for induction, etc.