The key difference is a complex array of social factors, not money.
I actually said the key difference is the quality of primary and secondary education, and within that I included basic values and work ethics as they are learned behaviours, not money. Money is just a facilitator.
IF a child's parents are intelligent and hardworking then are likely to be wealthier and to have more intelligent and hardworking offspring (intelligence is largely related to genetics and family environment/upbringing). Thus children of wealthy intelligent parents will likely go to university and do well, but that has nothing to do with wealth.
Without so much as Googling I am quite certain that we have no conclusions regarding the source of intelligence beyond "it's a mix of several things" that will include genetics, environment etc. To claim it's 'largely' genetic is a stretch. If you like you can post links to a few papers supporting your supposition but it's not accepted fact - we simply do not know enough.
Wealth in itself of course does not generate by some amazing chemical process successful children, but it does
enable parents to provide hugely powerful environmental and educational support. A product of a parent being wealthy is that they can ensure maximum exploitation of any spark in their children.
The converse is also going to hold, with children of lazy workshy parents being less likely to put in the effort to get the grades and go to university.
But what makes them less likely? I don't think you can conclude it's their chemistry.
If a child is bright then they will tend to do well regardless of the school system, especially if the parents are responsible. Having poor teaching standards wont limit a bright child. I know, my teachers were god awful and I had a terrible time at school due to various reasons. But I worked hard, I got some supporting books, often free from the school library. My parents were close to the poverty line when i went to university. Finances had nothing to do with it.
I agree that particularly bright individuals will do particularly well in spite of difficult circumstances, however that's the exceptional situation. The vast majority of people will fall well within the bell curve of results for the institution where they were educated.
These are generalizations and i am to saying this is always the case without exception. But it does explain why on average students come from wealthier backgrounds. There may be a small minority of children who get severely disadvantaged by the school, but again, a bright child with caring parents can take care of this without cost.
My parents were very supportive of my homework, helped teach things above an beyond the school curriculum, made sure I took the mock exams seriously.
Any parent who thinks their child is not being properly educated at school can do a lot to help with no or minimal costs.
I, too, was lucky to have very supporting parents. Almost undoubtedly I owe much to my parents for striving hard to ensure that I had the best start they could possibly manage to afford me. Many of my peers at school did not have parents like them and they are significantly worse-off as a result.
Think of this scenario: at birth you take two twins and give them to two different families. One you give to a poor distressed family living in a poor area with poor schools, the other to a wealthy intellectual family living in a cultured part of the country, and they attend top schools, have extra tuition and piano lessons.
After 10 or 20 years, which one do you think would have done the best? I would be willing to bet that 9 times out of 10 when you did the above, the twin given to the wealthy family is
significantly ahead of the other, and that wouldn't be because of genetics.
I do not think it is possible to argue that every child born in the UK is afforded equal opportunity to succeed.