I'm very glad he wasn't my dad. When his son was 11 he got a place at an outer London grammar school. Corbyn didn't want to accept that. He wanted his son to go to the nearest state school, which was not in a good way at the time. His wife (second of three) insisted that their son went to the better school, and this led directly to their divorce.
My parents, by contrast, were absolutely delighted when I won a scholarship to a direct grant school (independent but taking a large number of scholarship/assisted places girls with a government grant, abolished by Labour in the mid 70s) and supported me wholeheartedly. Having said that, I didn't come from an affluent middle-to-upper-middle class background, as Corbyn did, so it transformed my life in a way that perhaps it wouldn't his child's.
I have little time for Dianne Abbott, but when faced with an even more stark dilemma (to pay for her son to go to City of London Boys' or send him to a failing school) she put her son first. Harman did similar (St Olave's, an elite state grammar school) and so did Blair (London Oratory, elite Catholic 'comprehensive' school).
Now I'm sure Corbyn and many of his supporters would have said that many families have no choice in those circumstances so he was showing class solidarity by wanting to stick with the local schools even when they were not doing well. He may have felt, as many do, that a bright child from a supportive home would be OK in the long run. I'm not so sure about that. I would have hated being at a school with a constant turnover of teachers, hardly anyone of my level of ability, behaviour problems galore and no specialist teachers in some subjects, e.g. Science or Maths.