One of the important considerations in dealing with women's concern about modesty in child birth is the question of pikuach nefesh. While everyone seems to hold that child birth is pikuach nefesh - there is no consensus on what is permitted. Rav Moshe Feinstein permits her husband to go to the hospital with her on Shabbos as well as to travel to a more distant hospital - to avoid upsetting her.
In regard to birth couches (doulas) the Chassidic poskim allow a woman to call her coach on Shabbos and for the coach to take a cab to the hospital to assist and also for the coach to wear a beeper. In other words Chassidic poskim treat birth coaches as they would members of hatzola. It would follow from this fact that the Chassidic poskim should be more willing to permit female emts for child birth than a Litvishe posek. At this point I am only providing conjecture. But it would be ironic if the Chassidic poskim are more sensitive and concerned with female concerns.