Interesting one. I think some children who go to specialist/independent schools have access to resources and well trained/qualified staff who can help them close the gap (dyslexia focussed schools like Maple Hayes or Moon Hall or even normal indys who have a dedicated support department). Saying this, the potential for this to happen has to be there - which is not always easy to predict.
I was always frustrated that my DS made very slow progress at primary (1 year progress over 2 was usual but he was reading at age 7.5 at 11 years old so sometimes much bigger gaps opened up). He has multiple SEN like dyspraxia, dyslexia, ADD. Speech etc and has always been developmentally I would say 2-3 years behind peers (no global delay diagnosed though)
Having done home ed for almost 2 years we have achieved a similar rate working 1:1 with specialist programmes and materials like Toe by Toe and Nessy. It satisfied my curiosity to an extent as to whether I could do a better job focussing on his areas of need in a quiet environment.
At secondary he's gone to a specialist unit within mainstream where they work in small groups to the child's level. They are aiming for entry level quals rather than GCSE which I think is reasonable.
I had to weigh up whether he would do better at Maple Hays (we would have had to move though as too far to travel) and what suited his temperament, peer group etc. I'm pretty sure an academic focus would have been frustrating and things always just out of reach.
I think the SENCO was partly being honest about the state of SEN:
www.bbc.co.uk/news/education-58474416
but partly feels like she's giving up - as I think for some children it certainly is possible - does your child have an EHCP and are they saying then that they can't provide for needs to meet outcomes?
Support at mainstream secondary does seem somehow harder in practice to achieve - talking to SENCOs as we visited schools pre-Covid. Ours would have sunk without trace, of that I am sure!