I think the graph you shared that kicked off this mini-education discussion might need a little more examination, tbh. The weasel words there are "state schools". For HE stats like this the Universities, particularly the most exclusive ones for their own purposes, like to shine the best light on "state school" stats. They love nothing better than kids who have been put through privileged, private education from prep to GCSE who then sit their A levels at a state (often selective) school. They're classed as "state school" kids because of that L3 experience. After that they also love the state grammar school kids who make up way above their 5% national figure in the most exclusive HE intakes.
So we have to be wary of figures that show "state school" classification without any qualification and without any indication of socio-economic background. Even the HE stats that do attempt to factor in socio-economic data (deprivation) do so on the basis of mean figures for geographic units, not the kids themselves.
I think this chat started by looking at how Sunak's moderately wealthy, middle class parents had bought him a privileged education and it seems that such an ability and determination applies to many in immigrant demographics. This shouldn't really be a surprise as international immigrants do always represent sub-sets of their sending populations; by definition an immigrant must have the determination to seek a better life, be open-minded to making that in a distant place, have the ambition and guts to break from what they know and, in many cases, the economic where-with-all to make the move.
None of which changes my view that private schools and grammar schools should be banned.