The central issue here is really 'self-starters' v the rest, not state v independent. Self-starters - those with drive whether or not support is in place - are highly praised by many employers.
The independent sector will produce its fair share of self-starters and greater resource and support may well turbo-boost their performance but that core drive still needs to be there.
You could argue that contextual uni offers are, in part, designed to recognise the self-starter in a more challenging educational context who is 'only' achieving or predicted to achieve an A rather than an Astar but, in that context, it is understood the A is hard won and, to a larger extent, reliant on their own initiative.
One of the (many) reasons that larger employers have their own multi-stage tests is that academic degree performance is, of itself, not necessarily a good predictor of whether a potential employee will have the initiative or communication skills to thrive.
I think an elephant in the room is informal, behind the scenes tutoring that is more likely to be prevalent in some parts of the state system, whether for 11+, GCSE or even A-level. This blurs any attempt at a neat state v independent analysis or line of argument.