"they can save up and get a private midwife-of their choosing"
Yes. It'll definitely be something someone on benefits is able to do. Not. 
"I don't believe, that as mammals, women have ever crept off to have babies alone"
You might not believe it but it in all cultures at all points throughout history, a proportion of women have chosen to give birth alone.
From Sheila Kitzinger's 'Rediscovering Childbirth':
"Women of the tribal people in the Kalahari desert of Botswana and Namibia in southern Africa are exceptional in that with their second and subsequent births they do not expect help from other women during birth. Yet they, too, see birth as a transforming experience which draws on spiritual energy. They are proud of giving birth out of doors, in the bush, alone. While men dare death by going into trances and hunting antelope, women accept responsibility for birth. They believe that the only enemy is fear. Women aim to enter a "powerful altered state of consciousness, one in which great learning and personal growth are possible" (1). The solitary birth experience is seen as an important process in maturation, in becoming an adult able to be powerful, responsive and productive....In West Africa, among the Bariba of the People's Republic of Benin, unassisted, solitary birth is also the ideal."
There are other anthropological accounts of women birthing alone in Native American communities in the past.
"I expect that midwives are more understanding and compassionate when things go wrong because they are paid to be and it is their job"
I think it's got more to do with the fact that they spend a lot of time talking to women and listening to them. They value women's individual feelings and experiences and take their fears seriously. You don't. And you don't know anyone who's chosen to give birth without a health professional present so you feel free to stereotype and be judgemental.
"Imagine the sheer terror of going off alone and things going wrong"
Amazing really that women experience 'sheer terror' in hospital too when things go wrong. They should be feeling absolutely fine, shouldn't they, because we all know that DOCTORS CAN FIX EVERYTHING. 
Really exotic fruit - can you not take on board that women who give birth unassisted DON'T HAVE THE SAME FEELINGS OR EXPECTATIONS AS YOU DO.
Seriously - try to make a conceptual leap. Put yourself in someone else's shoes for a minute.