"Just incompetence and a desire not to be wrong." Projecting maybe... But I am extremely biased, my family is full of stubborn autistic men who just cannot admit when they are wrong!
Most people I know, me included, did NOT have the backing of the school. Autism is not always obvious in such a regimented, timetabled environment, it's usually only when a child isn't coping that teachers will even entertain the idea and even then there is a lot of push not to put a 'label' on it.
"I think my child has one or two mild autistic traits which were a challenge in the early years of primary school but not so much now. Does it meet the diagnostic criteria for autism?" Do you really think I can answer that with such little information? I assume that is deliberate as you have been deliberately vague throughout.
The assessment for an autism diagnosis is just to assess one thing, whether or not that child meets the criteria for an autism diagnosis. If it seems a bit narrow, it's because it is.
It sounds very much like you went into the process looking to be proved right, when you weren't rather than accept you might be wrong you started to pick holes. You think you are right and the assessors, your wife, your child's teachers are all wrong. Now put yourself in an outsider's shoes, would you believe you?