When I was birthing my DD in a Private Hospital, I actually wanted to lay on my back (mostly because I was just exhausted) and my midwifes & obstetrician told me this was not a great position, encouraging me into another. It was good because as a 21yr old first time Mum, I was really uninformed so their advice was invaluable. I ended up pushing in a sort of side on crunch position and didn't tear or anything so that was great. It might be worthwhile to explore some other health professionals until you find one who will support you in the way you are wishing for.