I can't say I approve of the term 'woo'. See below what the dictionary definitions are. Surely all that's being suggested is that the concept of a Doula is a bit new age or hippy, which is not quite the same as 'woo', based on 'woo woo' stemming from 'voodoo' Just think one should be careful language wise.
woo woo - Wiktionary
(slang) A person readily accepting supernatural, paranormal, occult, or pseudoscientific phenomena, or emotion-based beliefs and explanations. That reporter is a bit of a woo woo. (slang) Those beliefs. He is really into all that woo woo.
Urban Dictionary: woo - n.(or adj), the way a person is when they uncritically believe unsubstantiated or unfounded ideas. Short for "woo woo".
Cambridge Dictionary - woo-woo adjective /ˈwuː.wuː/ /ˈwuː.wuː/ mainly US informal, based on false beliefs or imaginary things, rather than reason or scientific knowledge.
The Doula I used was marvellous. Someone at the time described the concept of a Doula being a sort of lay midwife. There is some truth in that as a description especially where advice re planning is concerned (ie - advising the mum to be to 'make a birth plan' not saying what should be in it using medical terms of reference) and also in that many Doulas have trained to be Breastfeeding Counsellors or in other areas like this.
My Doula was on a committee at the hospital I was treated at which meant she was there when my case and the mistakes made in Delivery on the night of my son's birth came up to be discussed. Her involvement meant she could not stay for the actual discussion. I mention this mostly to show that Doulas need not be people outside or out of step with the medical profession.
On the night of my son's birth, my Doula was much more professional in her approach than the midwives and nurses I saw. One told me to shut up because I was ill and let the other people in the room answer for me. In the 21st Century that ought to be a sacking or suspension offence and I still find it quite unbelievable that someone would say that in front of witnesses.
As far as Doulas are concerned, if one is sufficiently intellectually engaged to know what a Doula is and contemplate employing one then one can choose carefully. They certainly have their role as a birth partner and can offer help and support pre and post nataly.