I suspect parents cant wait the years it takes nowadays, and trust, especially with girls can be an issue. Again though, it is an assessment for possible diagnosis, not a diagnosis they're applying for. (Well normally!)
TBF I 'shopped around' for an NHS assessment!
I got my DC into Gt. Ormond St hospital services, (with their help) for assessment, because trust in the local services independence had been destroyed by social services, after they took the schools lead that I must have caused my child's difficulties and visible differences.
Tiptoe walking, poor balance, and walking around flapping hands with wrists carried at shoulders, (hypermobile) academically ahead but huge misunderstandings over meanings of questions and the slowest writing, but generally socially disinterested at school - single very good friend outside - all so obvious looking back - but schools opinion that he didn't need an EHCP but 'less TV time' carried weight back then. Not actually having a TV was treated as an affront to their decision, and suspicious - families like mine were supposed to have TV's and dump kids in front of them! Not arts and crafts, reading, and outdoors. We were in the wrong lane in their eyes.
SS told me that schools and their opinion, would influence any assessment now, so I bypassed them and appealed to the best NHS service I knew off that I believed had experts who would look at my child without involving politics, back covering, or prejudice, and whose decision I could trust even if they decided parenting was a factor.
I wanted my child to be put first, not the adults arguing about him. Wherever they go, I think that may be were many parents are coming from.
We ended up with a firm diagnosis, and the information that the way I was parenting was what had carried him through as far as it had, and I needed to keep doing it and I'd been right to request assessment for equivalent of an EHCP. (refused) The majority of meltdowns in school were being caused by bullying (they observed it repeatedly over school visits - inc lessons, assembly, and break.) and how it wasn't being handled, and the reason we weren't seeing meltdowns at home, wasn't as school claimed that he was too afraid to, but because the home environment had routines and consistency of decisions and he could make predictions confidently, which worked well for him as well as the NT others. (They then looked at Dad, Grandma et al, and out the family history and fears tumbled)
I think most parents aren't refusing an NHS assessment, I think they're refusing to wait the length of time now required for one, or trying to bypass poorer local services.