Having your water broken to start things that are not already started is a BAD IDEA. It will generally lead to lots and lots of interventions if the baby is not ready to come out. It can also cause cord prolapse, and I for one would avoid it like the plague. If your baby is not ready, and you attempt to force things, your chances of ending up with a bad experience are greater than if you just leave things alone and then insist on an intervention-free birth at the hospital, a few days later.
Yes, you could end up with an insensitive doctor. Hell, you could get in a car accident this morning and end up with one anyway---we all could, it's just a risk of living. AND it is a risk of birthing. You could also end up with a superb, very able and nice doctor. Remember that. Also remember that your chances of ending up in a very stressful situation, and being ultimately sent to the hospital, are much greater if you try to use the water breaking as a method of induction. What I mean is: Personally, I would be totally stressed out if my medwife did an intervention that didn't work and I ended up in the hospital AND with the pressure of if-you-don't-deliver-within-24-hours-after-water-breaking-we-cut-you type of thing. Under those circumstances, any doctor that walked into the room would stress me out and so would everything else. BUT if I ended up in the hospital at 43 weeks, having gone peacefully into labor on my own, on my baby's timeframe, I'd be less stressed to begin with, and in a better frame of mind to handle whatever the heck got thrown my way. I would also NOT have the pressure of them watching the clock because of AROM (artificial rupture of membranes.) I would also sleep easier afterwards, because no matter what happened during the labor, even unlikely emergency stuff, I would know that I at least did not let anyone do things to me that caused the problems. Anything that happened would be an act of nature and not because of interventions. Does that make sense? There is nothing worse than feeling remorse after a birth, like "if only I hadn't done XYZ maybe things would have happened differently." I speak from experience on that one!
If it were me, I would totally trust my baby. I would believe that he/she has a good reason to be born at whatever date, and I would also believe that babies choose whom they want at their births. I also believe, I really do, that if a baby does not feel safe with a certain care provider or place, they wait until they do feel safe. Such was the case with my second. I had medwives, and my son would NOT let me go into labor until 42 weeks, after I switched to good care providers a couple of days before. Maybe your baby has a good reason to seek a hospital setting. Maybe he/she knows that some assistance will be required or just wants to play it safe! Not that they understand it in a conscious way, but I truly believe that babies understand what to do on an unconscious level.
Sorry for the novel. Just my two cents. Good luck and please trust your baby above all else. Certainly have more faith in the little one, and in your body, than in artificial means to start labor.