I'd be extremely wary of asserting that privately educated kids are any more successful at university than state-educated kids (on the same course) when there's no evidence to bear this out.

If, however, you want to convince yourself that the amount of money you've spent on your child's education means they're smarter than the rest, go right ahead and believe that.

Universities don't select for whether a candidate has "reached their full potential". They select for what that potential is.

It depends on how you measure it. A kid from independent school with AAA is going to be worse than a state school AAA kid. But state school AAA kid might be 3-A* kid in independent school and might do better.

For smarter. My narrative is more prepared, used to working harder. And also, it's self-selection. If you are not made for that type of education, you are going to leave.

No matter if universities select for potential, or operating near it. Both would be nice. Now, it is increasingly also how they will shape the society if they accept student A instead of student B. And they want to shape society in a way that discriminates against certain kids for who their parents are.