>> We do get factory worker's children in the main admission, proof that it's possible.
I don't doubt that it is possible. What I'm saying is that it is not common, nor easy. Overwhelmingly education favours the kids form the more privileged backgrounds, who can afford not only a good education, but also all the expenses of actually studying for a good education (for instance, not having to work part-time to support their studies) and who are better placed to take advantage of a good education once they graduate at whatever level (e.g. because they come from a good family and people will take care of them).
Also, France is a bit special in that higher education is cheap or even free. This is not the case in other industrialised countries, notably the USA and the UK. In the UK in particular, studying for a higher education qualification often means having to repay a student loan debt for a very long time (like in my case.
>> Life's what you make it, and society can't offset your own will.
In principle, perhaps. In practice income affects every aspect of your life. It affects the choices that are available to you, and it affects the outcomes of those choices.
For instance, if you go to a good school where good kids don't do drugs you'll be much less likely to do drugs and harm your prospects than if you go to a school in a poor neighbourhood where that thing is far more common everywhere you look. Same goes for (violent) crime and so on.
Also, it's the case that when you're poor you can't afford to fuck up even a little bit (not to mention, a whole lot). When you're rich it's much easier to get off lightly, or even get rewarded for screwing up. I have personal examples of that also, but others have discussed it. I'm looking for a source, but I think Noam Chomsky has talked about this a bit.