Is he's happy and settled, I'd stay private, especially if current school are meeting his needs.
As another poster said, it's hard to predict the future - I'm guessing your DC doesn't have an EHCP?
I have a 12yr old DD who is neurodiverse, suspected ADD too as she has a lot of the traits. I've been a PITA and the HT's worst nightmare right from Yr1 as I was constantly fighting the school to get her assessed.
Changed schools for my DD to a local village one when she was in YR3, her first primary kept batting me away as 'nothing will happen until Yr 4, she won't get an EHCP'. Soon as I moved her, (which I agonised over) her new school phoned me within 10 days over 'concerns' - Excellent, this is what I was hoping for, got the ball rolling and she had EHCP within 12 months, had to fight for that.
Having the EHCP means I cause merry fucking hell if her TA is used for other children (this happened but stopped straight away) and also ensures she had adequate support at school.
Mine is Yr8 now and is not a social kid, she struggles to make friends and doest know what to say in conversations with peers her own age, she's a summer born too.
We visited oh so many secondary schools, looked at one SEN secondary and came out a bit deflated, 90% of the students were boys and I wasn't putting my daughter in a school with that ratio of sexes. We opted in the end for a out of catchment secondary but the SEN department have been absolutely fantastic with her and she thrived in Yr7.
We looked at our local secondary as it was supposed to have a good SEN provision, I could never get hold of the SENCO, I had to be a pain in the arse to even get a meeting with her, but in the end we decided we needed the communication and peice of mind to know if I need to speak to staff I need to be able to get hold of them and it was the best decision for us, having the EHCP has given me the clout to do it though.
Sorry for the essay but if my DD didn't have the EHCP, I would have had to find the money for private school, we have a good non selective one half a mile from home which a lot of local kids seem to start with wearing the local comp uniform but it doesn't go unnoticed that within 1 year or two they are wearing the private school one. State provision is so stretched, I doubt your son would get a look in without an EHCP and even if he did the fight is bloody wearing.