Really interesting thread. We looked at private primary for DS, due to over-subscribed state schools locally (SW London), but ultimately he got a place in the local school. They have been amazing in supporting him and us through his ASD diagnosis.
He gets full-time 1:1 support, as well as input from EP and support from autism outreach service and behaviour service. We are currently going through the new version of "statementing", which is required to ensure that this level of support can continue.
School is 3 form entry, and has tradition of strong pastoral care. Academics are fine and improving under the current head. I worried about it being too big before DS started there, but have been proven wrong - DS has hugely benefited from the greater budget flexibility that a bigger school has. They have a full-time (brilliant) SENCO.
Honestly, though, while a small number of DC go from the school to highly selective state or private schools, that is the minority. So, if you are set on a minor public school, I am not sure it would be the right choice. Personally, I would weight most highly where I thought my DS would be happy, as that has so far been the most significant factor in his development and learning. You will know what matters most to your own DS - they are all so different.