This can be a very polarizing topic, but here are my thoughts.
I had midwife attended birth center deliveries for both of mine. But I was low risk and I had relatively easy (although quite long for the first) labor and deliveries. The midwife run birth center I chose was affiliated with an OB at a nearby hospital, and on a block that was perpendicular with another hospital. If anything were to go wrong, we'd have several options on getting additional assistance. Our midwives were CNMs. IDK what other kinds of MWs are out there. Pre and post natal care were amazing. I'm talking at least 30 minutes each visit with a midwife (there were 3 in the practice at the time), very dedicated to us and our baby, never felt rushed... it helped make the pregnancy a really amazing time for us.
I would have been more hesitant to choose this route had it not been for all the back-up potential built in. Looking back I think it was the perfect choice. I have a friend in a nearby state that had a midwife attended birth at a birth center within a hospital. If your wife had a complication free, drug free delivery the first time around, I would really consider it. But she could potentially have all those things in a hospital, too. So I think this decision comes down to personal choice in care and services... I wouldn't hinge it on the cost at all. It was not a factor in our decision as our co-pay was the same for the prenatal care. I may have paid a few hundred more for a hospital stay had I gone that route, but ending up at the hospital overnight was a possibility, however remote, with the MW run birth center, anyway.
You will get a ton of anecdotes that will make you lean one way or the other "Emergency c-section, unavoidable, had I not been in a hospital w/ an OB, baby and I would be dead!" or "Cascade of interventions led to the worst birth experience ever and preemie that is still struggling w/ xyz due to not baking long enough!" I will say, I have never personally heard of a nightmare birth center/ MW attended birth story, though. Like, people are glad to be in a hospital, or so sorry they were at a hospital. But I don't know a single birth center mom that regretted their choice. I'm sure there is someone out there, though, I just don't know of anyone. It just happens that in my circle of friends, about 50% are birth center moms and almost all would do it again that way. The only mom that chose differently for her subsequent 3 kids was a mom that was to be a birth center mom, but was told she had preeclampsia, ended up getting induced at the hospital. Baby got an infection there and was on antibiotics, later had gastro-intestinal issues that she felt was related to the antibiotics that wouldn't have been needed had she not been in a hospital, etc. So for her next 3 she had home births with a CPM (cert. professional midwife, no nursing background). She's been very happy with her prenatal care and deliveries, but she's the kind of mom that wanted zero ultrasounds and would rather listen to her gut instinct than a medical professional. I personally cannot rely on my gut that much, it steers me wrong all the time.
Best of luck on your decision, I do really think this is one to be made w/o the price tag in mind. And congrats on your new LO!