Thanks so much for this helpful response. I really appreciate the specific info, and feel more hopeful!
I also talked to midwife Georgia Rose on the phone yesterday. She had very helpful info, which I'll post here for the benefit of anyone searching this same question later.
She says that in general, private-practice midwives do not accept PCAP/Medicaid, because the reimbursement is simply to low for them to afford. At the low volume at which private midwives operate, it's just not feasible.
She said in general, if you're on Medicaid, clinics which offer midwifery are the way to go, because they operate at a high enough volume to be able to accept Medicaid reimbursements. The downside is that it's not a one-on-one relationship with a midwife. However, even in many private practices, you are not guaranteed to have "your" midwife attend your delivery.
She gave high praise to the Morris Heights Health Center's Birthing Center in the Bronx. I've also read lots of good comments about this place.
She also mentioned the Brooklyn Birthing Center and Bellevue.
I will post more info as I find it.
Meanwhile I'm still curious about anyone else's experiences with this.