Some people do nothing and have everything, that can't be right
I agree 1 trillion percent! It drives me nuts that there are people in government who, frankly, I wouldn't trust to look after a cat, but get paid hugely more than say, hard working teachers, or cleaners all because they went to Eaton - but it's life.
The only way to make it fairer is to ban the right for people to inherit wealth. Literally a ban on inheriting significant assets above say £10,000.
Another way is to get rid of all private and grammar schools.
Since some people find the idea of nationalising the railways drastic I don't think either of those measures are likely to happen anytime soon.
Other than something extremely drastic like that, getting rid of the Royal Family will accomplish nothing.
Another thing you fail to consider and to which I have had my eyes opened recently, is the role of nepotistic influence. I have a friend with huge family who has literally walked into every job and received an offer at every university through contacts. One job she has applied for recently she didn't even do the test for, it was done for her, because she has two cousins working there. Had I applied at the same time as her, I wouldn't have had a lookin. She got confirmation of the job today.
So how do you deal with things like that? How do you deal with say, some people getting great work experience opportunities because of who they/mum/dad/great grandfather/15th cousin twice removed is and some not?
If you want a fairer society you need a revolution. A big one. And no one wants that. It's been done already, and when it all dies down it's business as usual, the same thing in a different guise.