In case it is of any use to you OP … I will share two riding students, both about 10 yo as I know them, who very much reflect some of the personality and behavior you are describing for your daughter. Based on paralyzing shyness each girl has great difficulty coping with less familiar situations, and especially with speaking up – whatever the reasons behind it, which I don’t know.
One girl that I taught for 18 months was definitely ‘on the spectrum’ and was far more interested in the barn cat than she was in the horses, or riding. She related to all animals, but not particularly to horses. I had several in-depth conversations with her parents over the months, as we agreed on parameters and direction.
Her parents told me that riding lessons were their idea, in an attempt to expand and grow her self-imposed horizons. She told me the same, that they were making her take lessons. I thought about backing out of instructing her at that point, if she were not a willing student. But in the end I decided that her parents knew best what was needed to help this child (10 yo at the time) reach an adulthood that she could cope with on her own. They are raising her. I was seeing her once a week for an hour. I decided to go with their judgment, not mine.
With the agreement of the parents, I let the girl guide much of the boundaries and limits of what she was willing to do – for months, walk on the longe line on the tamest horse. She very gradually became confident enough to ride at the walk on her own, steering around trees and obstacles. She never trotted - we tried, and she went into PTSD meltdown panic. I said I wasn’t pushing her on that, we’ll do what she agrees to do. She gets to set some boundaries. That also helped her confidence to try other things. One thing she did expand on over time was riding other quiet horses. At a quiet walk, of course.
During the early months she said almost nothing during her lesson time. One day she just started talking. After that, with light encouragement, I heard all about her schnauzer, her little brother (a pest of course), and a little about the family trips to Canada where they have family. She had a clever sense of humor, too.
For me, the triumph graduation was a couple of lessons where I rode a horse alongside her and her horse on a very tame trail ride around one grassy paddock, with trees to navigate. She was functioning on her own and I was saying little about how-tos. Her parents finally felt that she was at a finishing point for riding horses. She’s moved on to other activities that she has a maturing interest in.
This girl who was 10 when we started and 11 when we finished, was diagnosed autistic, don’t know at what level. When she started was barely able to function among people outside of her family. She was gaining interpersonal skill over time. I doubt she will ever be outgoing.
For whatever it is worth, she had genius level intelligence. She once read off a complex legal liability paragraph and explained what it meant, with no prodding – guessing many adults would not have understood it. Age 11 at the time. Her dad told me she was in advanced classes, got A’s on everything, quickly and perfectly completed all of her homework and exams with no problem. I think that someday, as an adult if she stays on a good developmental path, she’ll be able to live and function on her own fairly well. And be employable at a generous level.
The other child is one I’ve taught a few times but is not my regular student. She’s 10 years old and behaves very much as you describe your daughter behaving, OP. She’s only been in the program for a few weeks. The girl has a lovely sweet smile, but is almost unable to speak to people she doesn’t know well. She is in a one-on-one focused private lesson program. It is very much what she needs with an instructor who is both sensitive and pragmatic. The girls gets support with everything she needs, as she needs it. But her instructor also stands back and lets her do the things she has learned to master. Her instructor is attentive to the girl’s situation with the tack and the horse, with a matter-of-fact attitude that seems to give the girl confidence.
On the one hand, I don’t think the girl really wants to ride horses. Shows no enthusiasm that anyone on the program side can detect. This is her mom’s insistence. Mom restrains her own domineering affect with difficulty.
On the other hand, the girl does best when she is on the horse! 
She could be quite a good rider – other than she isn’t that keen. It’s hard to get her to pick up the reins and steer! Constant encouragement to sit up in the saddle and actively ride. But once she does do it, she does it well and get good results. She is quickly mastering the trot – when she actually makes the effort.
It is harder to encourage a rider to progress when they are truly fearful on a horse. Progress often means faster speeds, even at the walk. And that increased motion and speed is what triggers some people. Even if they are drawn to the idea of riding – but not so much the reality. Showing them how their skills will maintain control, helps. But in the end, they don’t really want to be there.
No statistics to support that, just general observation. They can be very good riders, good at horse husbandry, and some succeed at the highest sport levels.
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