Wow, I would have had the same expectations you had, that the midwife would be there to offer a lot of labor support.
I didn't know birth centers did inductions? How did they do it?
As for the induction, they tried other methods before chemical induction. I believe they suggested I start homeopathic induction, sex, evening primrose oil (vaginally), and nipple stimulation around 40.5 weeks. Then they inserted a foley catheter at about 41 weeks. The plan was to try gel induction at 41.5 weeks (outpatient basis at the nearby hospital) and, if that didn't work, I would become a hospital patient at 42 weeks and get Pitocin. The gel worked (sort of), so I went back to the birth center until labor stalled at 9 cm.
This time, I made sure our provider is OK with going to 43 weeks before we even tried to conceive.