In the end, I think it is the parents and the upbringing that helps the children in their future, regardless of income.
We never had much money even though we always worked, but we had enough to do the best we could for our 2 sons. Our local state school was appalling, so we applied to the local catholic primary school, being open about the fact we were not catholic. The fees were low, the school was small, and the ethos seemed excellent. They went on to the catholic co-ed high school:-
www.spcc.qld.edu.au/services.php?id=59&category=about
It all worked, the lads did well, and got their uni degrees etc. One teaches now at the same high school.
BUT, I think the fact we were not well off helped rather than hindered. They learned that they could not have everything they wanted - someone had to work hard to get things. They had to do their share of housework - all 4 or us shared it - because both of us parents worked. They saw how hard it was for their Dad in his working life,without any academic qualifications at the time (we both went to university later, when they were grown up). I suppose we had a 'middle class' (for want of a better term) attitude, without the financial comfort to go with it. We analysed things , discussed the news, discussed the world, as we sat around the meal table. We expected respect from them, and good manners. (THAT was an uphill struggle sometimes) They knew that although their choice of subjects was theirs, going to university was a given.
So - perhaps 'class' is just an attitude.