Speaking personally as a working class person with a DH, both of whom had very few qualifications on leaving school. I work in an average wage job, DH in a Min Wage job. IQ tests put DH and I over 135. DS's have not to the best of my knowledge been tested.
We have 3 DS's who were educated after primary school with a bequest from my DGM who bought a house in London in the 1960's, which was sold in 2001, with the will proviso offering private education at secondary level, offered to my DC's with the sale of the house, or we could live in the house to be sold when the DC's were over 21yo and the money distributed around various relatives. We took the first option, with agreement from the relatives. They were old at the time, and likely to have passed away if we stopped living in it.
My DS's are now an officer in the British Army, a Civil Engineer, and a Published Composer.
None of which would be possible, or was expected when they left primary school as they were labelled as being 1. Overly excited about sport, 2. Overly obsessed with maths actually a maths genius and 3. Over emotional about music and drama.
Private all the way in my opinion. They are supportive, and encouraging of the kid who doesn't fit the mold.