It isn't that the private school kids don't need it - the cognitive testing to secure that is scaled and peer-reviewed. It's that the state school kids by definition have fewer affluent parents, so fewer who can afford the testing.
It's not state or private: that's a proxy for affluent and non-affluent.
SEN is spread amongst the population pretty evenly. But if you can't afford the testing to prove it, you can't prove it. Same applies to EHCP application and provision - state reports carefully sidestep identifying the needs in full, all too often, and rarely make any recommendations that actually cost anything. Good private assessments will be more detailed and then come up with recommendations based on needs. There's actually legal aid for private reports for very low income families who appeal, so it's the lower end of normal parents who can't get any at all, unless they are lucky enough to get charitable support for the funding of them.
The SEN system is a mess. But it isn't that the kids of more affluent parents are privileged - bluntly, any kids with SEN are less privileged, by definition: the gap in employability and life expectancy for autistic people is much, much wider than that based on class for the typically developing, and if anyone thinks the SEN system works well for the more affluent, I have a bridge to sell them. It's the intersection of class and disability that is truly toxic.
SEN kids from lower income homes (which is not all kids in state schools - some attending the top notch state schools are better off than many in smaller, less glam private schools where I live, as the house prices in catchment are so obscene) are doubly disadvantaged. But all kids with SEN are, and the crisis in SEN provision makes the NHS look luxe.
I wish people cared as much about that as they do hating private schools, frankly. That much attention, focus and energy, and their life chances would transform overnight.