FWIW I think it's possible that Maddie and Caleb chose the birthing center - 2 hours away from home - because of its association with UNC. The midwives at that center are affiliated with UNC and have a working relationship with this major hospital. While they didn't know abut FATCO, they did know there might be a problem, and therefore they were close to UNC just in case. I think that's a reasonable responsible decision.
While deliveries at UNC, could also be unmedicated, it makes sense that Maddie felt more comfortable with the center. It's quite important for a woman, who wants a low intervention birth, to be in a comfortable and familiar setting. I didn't see anything wrong with what the midwives did or said - with the exception of them telling Maddie to go home at 4-5 centimeters. And I'm not sure she was in fact told that? I can't recall - and it might be that editing was involved.
For the midwife that said "sometimes babies wait" - this is actually true due to the fear tension pain cycle. No, moms cannot "hold a baby in." But there is something to be said about a woman who is so stressed, tense and scared, that the labor does slow down. It's not the baby doing it of course - but it is an evolutionary part of the process. Back in the day - you don't give birth with a saber tooth tiger around. You move to safer ground.