Being on arms & knees lets the tailbone swing out instead of pinch in, like when she is on her back, or semi-sitting, but squatting, or standing with one foot up on a chair opens up the pelvis best. I had one birth where the baby reached a point, just as you described, where caput was increasing, but no progress on the head descending. Finally I reached in and held the pubic bones apart as she pushed. She pushed for 6 hours altogether, baby had quite a cone head, but was born naturally at home. The Next time around, we were ready to help her as I had before, but she only pushed 15 minutes and baby came flying out! Quite a difference in time and effort, only a few ounces of difference in weight.
It sounds like none of these techniques were used, so I would think if she were to hire a competent HCP next time, she'll do just fine.