From personal experience in the workplace, and from family and friends who've worked in The City, corporations and served in the armed forces.
Is that good enough for you?
No.
The point I tried to make is that the leg-up comes from university. I also worked in The City etc and have family in the forces. The trajectory is set from transition through 6th form. I have experience of 'you went there?' from uni, but no-one gives a shit where I went to school.
I do, because friends from uni who did not have that boost had to work harder and I respect them for it.
There is a benefit, possibly quite significant, from public school education - but I have never seen any evidence that it is 'Old Boy Network' in the way that you appear to suggest.
Public Schools use university entrance stats as marketing - they will try to optimize students to courses to get the best stats that they can. Unusually, that does work for the kids, somewhat unlike the government's current stats regime.
And yes, not the sharpest knives can get through with some help - but I don't believe it is connection based as much as targeted tuition and coaching that boosts them.
At some level there is little difference - the school knows what the next level want, and tries to deliver it. But its not Mates' Rates, nor I Have That Tie.