It's hardly just luck though, is it? There seems to be this idea that those who end up wealthy only do so because their parents were before them, etc. Although there are probably some to whom this applies, I think we do a massive disservice to rubbish others' efforts in such a manner. All because you're rich does not mean you can simply walk into a top job without any qualifications/gumption whatsoever. Yes, the career I am going into means that I will end up as a high earner and it's rather disappointing that such an assumption will be made simply on the basis of my paycheck as opposed to what I have personally done to get there, and the same applies to many others - it's just that you are not personally privy to whatever sacrifices they have made to get to where they do.
Unless you are willingly sending your children to the most underperforming school you can find, living in the most ghetto-ised area you can, you yourself are applying/specifically advising your children to apply for jobs that are on the lower scale of economic wealth, your relative wealth (and note the word "relative") is working to your advantage. All because there are people out there better off than you are does not mean that you don't indulge in similar behaviour, albeit on a different scale when it comes to the number of 000s in your bank account.
You don't have to walk into an internship straight after university. Sure, it would be great, but it's not a necessity. There are many who start off in lowly paid jobs in the same or a different industry and, with a bit of luck, investigation, lateral thinking/education and sheer bravado (amongst many other things, which do not necessarily include spare cash to flash), work their way up accordingly. Not everyone does it, but from the position I came from, I truly believe that, in this country, more people can than you all realise.
However, from what I've seen on these boards, success is not something to shout about. Most people would like to brush aside people's experiences with the general phrases of "luck" and "private schooling" rather than realise that those are not the things/issues to be focusing on to encourage social mobility. You do not need money to get into certain careers - it may make things easier, but it is not a necessity. At the risk of sounding flippant, it's the same as a luxury car - it may make driving smoother or a more luxurious experience, but a bus fare would still get you from A to B just fine.
Anyway, I'm starting to sound disgustingly syrupy, so I'm going to stop now.