TL;DR: wealth helps a lot, but not in the "parents paid the diploma" direct way. More because good social status helps putting people in the right track. Hard work is still required IMHO (at least for the engineering track, I do not know about the business schools variant).
Wealth definitely helps in the sense it gives you a better environment to support you during your studies. About it helping you being tutored by fancy private organisms, this is true but I think it applies more to the "lower" part of the Grandes Écoles & co pyramid. You are not going to enter Polytechnique or ENS solely because of that. It still requires a lot of work or talent.
About the set of high schools/classes préparatoires funneling a large part of candidates to top schools, that is definitely true. But again it requires a lot of work, you are not going to move through Henri IV or whatever to ENS only because your parents have money. But receiving a good education from birth and having family and friends who knows how the system work, because they moved through it themselves, helps a lot.
But wealth helps because these places are usually in large cities (and mostly in Paris), which are expensive to live in. And to get there you have to come from good/prestigious high schools and have good grades, which again living in a wealthy family helps a lot.