I have ASD and find the whole experience of haircutting very stressful. I'm not sure I can fully explain it, but it just feels like massive sensory overload with all the noise of the salon, radio, dryers, cutting sounds, the feel of someone's hands on my head, the smell of all the products, the small talk and weird conversations, the other customers, the lights. As a child I used to have an instant meltdown, so my mum then friends cut my hair for ages. I've now found a hairdresser who doesn't chat, doesn't insist on washing, blowdrying or spraying anything, and lets me go in before proper opening time. I can cope with that, but still it's not easy and I only go twice a year.
There are comments on the thread that show a lack of understanding of autism and the social, communication and sensory problems it involves, especially the talk of 'desensitising'. When an autistic persons seems to learn to cope with social or sensory overload, it's because they are good at faking. I do this - I could probably get through a whole fancy cut/blowdry without you even knowing I'm autistic. But then I'd go home, curl up in a ball in a dark room and be unable to speak for several hours, and feel exhausted and extremely anxious for days afterwards.
We can't lose our sensitivity, because that's part of our autism, it's what we are. So if an autistic child is successfully 'desensitised', they've just learned to copy behaviour that makes little sense to them, and hide the stress, anxiety, and discomfort they are feeling. With me, and several of my autistic friends, this has led to mental breakdowns and poor mental health throughout our lives. So it's really upsetting to read the comments which seem to think outer behaviour is more important than helping a child to feel comfortable and taking account of their needs.
So YANBU to the OP. Your friend is ridiculously narrow minded if she thinks hairdressing is an essential part of anyone's childhood, and especially an autistic child.