Many here have pushed saying your daughter should be doing self-advocacy.

From what I have heard so far about your daughter, she is struggling enough with acceptance of her disability and is not ready for self-advocacy.
Children go through a process somewhere between their 8th birthday and their 10th birthday with their first initial acceptance and understanding of their disability. The brain does a shift between 8 1/2 and 9 1/2 that cause children to see the world differently.
I think you badly misunderstood what I wrote about self-advocacy. If we were in a room talking about it, I think we would have more in common than your

sounds like.
I am not pushing or suggesting pushing the child into self-advocacy. For example, I was not talking about making the daughter go to Guest Relations and have to explain things herself. I was talking about starting to talk to the child about feelings and trying to help her understand a little more about what she could do to deal with them.
Self-advocacy is a long road and I was talking about taking a few baby steps onto that road, not pushing the child onto the super highway of life.
I have a feeling that we are seeing self-advocacy in very different ways.
I think of self-advocacy as knowing what you need and doing something to get that need met. It's not vital to accept or understand the disability or even the need; just to understand you need something or don't like something and can do something about it. Many adults don't understand or accept their disability, but do understand they have some needs (hence, a lot of threads where people are writing "I don't accept that I have any mobility problems, but I have decided I really need to rent an
ECV this trip because I don't want to be in as much pain as on my last trip). If people waited until they understood and accepted their disability, most people would not get their needs related to that disability met.
Babies are born with some built in self-advocacy on a very primitive level. It's a survival mechanism, but self-advocacy later on is built on that. They don't know what they need, just that they are uncomfortable and crying will get their need met. As time goes on and they get more experience, they begin to understand their needs and that they can do something to get the needs met. It's not socially acceptable for a 4 yr old to cry to get their needs met; somewhere along the way, they get self-advocacy tools to meet the needs in more socially acceptable, age appropriate and effective ways.
I'm talking about the same things that you have mentioned - the child who feels frustrated and/or angry and goes to a quiet place (taking themself out of the situation that is making them uncomfortable - actually a Time Out on a pure, not punative level).
The child knowing they feel uncomfortable and going to that 'safe place' is self-advocacy; knowing they have a need and doing something about it. Some kids with autism may never get past that point and may never fully understand or accept their disability. To my mind, that's not important. What is important is that they have some tools to help them deal with what they need. That can come before they have any idea of what their disability is or what effect it has on others. This is dealing with self and self needs and is the first steps for being a self advocate. That may not be what you are thinking of as self advocacy, but it is what I was thinking.
In giving advice for that the child I was going on some of the things the mom had written:
My dd is so much more aware now that I have to plan ahead. Before she didn't seem to pay attention, now she does.
She gets embarrassed when I explain about her.
She is 8 years old and so badly wants to be independent but is not safe being independent yet..
The mom had also mentioned in another thread that her daughter had high functioning autism and tends worry and overplan things and get herself worked up.
This sounds very much like a child who has started to notice her disability. She is dealing with her discomfort in ways that are dangerous or not socially acceptable (running away, hitting people).
My thought was that it might work well to begin to use her want to be independent in a positive way to help her start understanding and dealing with her disability (baby steps). Independence used in the ways the girl is doing (running away) is dangerous. Funneling the independence can turn a negative into a positive strength.
Also, to use her tendency to overplan and get herself worked up in a positive way to help her plan on ways to help herself (baby steps of self-advocacy).
Kids differ a lot in the age when they begin to notice they are different in some way from their peers. They may not know that what is making them different is a specific thing called a disability that has a certain name. Some may begin realizing this by the time they are 6; others may still be clueless at 12. Kids notice physical differences at a very early age. I've had kids hardly old enough to talk come up to DD's wheelchair and ask "Why she have that?" pointing at the wheelchair.
My own DD has cerebral palsy, ADD and Obsessive Compulsive tendencies. It's probably easier with physical disabilities, but she already realized as a toddler that other 2 yr olds at her home day care could do things she couldn't. It came out with throwing blocks and crying when the other kids played with blocks. We figured out much later that the reason she did those things was she knew they could play with the blocks and she was not able. By the time she got to Kindergarten, she had noticed things like, the other kids had regular pencils and she had a stubby one with a grip on it and the other kid's worksheets looked different than hers. She knew this was different and she didn't like it one bit. Her way of dealing with it was to throw the paper and pencils.
I said in another thread that my DD was not able to self advocate. This is not completely true. She is not able to talk and does not like to use her communication device. That does make it difficult to self advocate.
One very memorable self advocacy occurred when she was 7 or 8. She attended an after school day care program 3 days a week and there was one male staff member who talked to her like a baby (the high sing-song voice and kind of simple words people use to talk to a baby). I had heard him and thought I took care of it by telling him "She can understand you. Please talk to her like the other kids."
Then one night when I picked DD up, she started signing "bird" to me. That led to a 20 questions trying to figure out what she wanted to say about "bird"; had one been to day care? had she seen one in school? had they talked about birds? was she talking about birds in general, or a specific type of bird?
We finally got it narrowed down to a specific bird, "turkey" and started the next 20 questions to figure out what she wanted to say about turkey. It didn't take long until she pointed to the male staff member. I felt like a light bulb went off in my head

and said "You want to say that he is a turkey". She practically nodded her head off, then added the sign for baby. "You want to say he is a turkey because he talks to you like a baby".
Bingo! That was it. And, she started to push me toward him with her wheelchair. She wanted ME to tell him that. I said I would only do it if she came and told him herself with her signs, with me translating. That's exactly what we did. He was very apologetic and said he did not realize how much it upset her and that he would try not to do it again. He also told her to tell him if he acted like a turkey again. She did not necessarily understand or accept her disabilities, but if we had not started with baby steps as a toddler, she would not have been ready to advocate for herself as a 7 or 8 year old.
I know someone who is almost 19 now and has Aspergers. People had noticed him doing 'weird' things since he was little - for example, he would sit or lay
under the bench at church, even when he was 6. He was diagnosed with Aspergers when he was about 11. He still doesn't completely understand his differences; he's still working on it. But, he did know when he was 6 that none of the other 6 yr olds were sitting under the bench at church. Now he knows that he did it because of his Asperger's and that it was a 'safe' feeling place for him with all the noise, the music and the minister facing him and looking at him. At the time, he thought he felt that way because he was sinful.
He knew he was different, but partly because of his parent's denial, he didn't get a diagnosis until he was about 11. He was relieved to find out that all the differences he had seen between himself and the other kids his age had a name and en explanation. Does he understand and accept his disability, even at 19. No, not totally. But he does have tools to deal with it and people he can call to help him self advocate when he gets into trouble (like when he locked his keys in his car and had to think thru what to do).
I've seen a book called
All Cats Have Asperger Syndrome that is written for kids and would have probably helped him a lot to understand his disability. I'd highly recommend that book to the OP.