Brown did not have an "extremely privileged background" at all - his dad was a Church of Scotland minister. Having known a couple of ministers' sons, I can assure you that although they often get a nice manse while they are employed at the local church, they are not rolling in cash by any means.
Clare Short came from a working class background and worked her way up through university and student grants that her colleagues all got and later abolished for others.
John Reid's parents were a postie and a process worker. David Blunkett's dad was killed in an awful industrial accident (for which no compensation was paid bc he was over retirement age) and grew up poor. Robin Cook's dad was a chemistry teacher which wasn't growing up in squalor but it's hardly an extremely privileged background.
Alan Milburn, Margarett Beckett, Jack Straw were all from working class backgrounds and went to state schools (Straw's was a state-funded grammar school, if that makes a difference).
I'm not saying that their working class credentials made them better politicians - Jack flipping Straw! - and neither am I saying that the Labour cabinet did not have people that did come from privileged backgrounds - Patricia Hewitt! - , but it's just factually not true that they most were "close to being toffs" and all from privileged backgrounds.
It's also not true that Labour used class as a weapon to beat the Tories with for thirteen years. It's simply a fiction. It's almost as if you have entirely missed out on the rise and reign of Blairism (lucky you...). Blair's government was one that was petrified of the "C" word and had Mandelson out telling the world "we are intensely relaxed about people getting filthy rich".
Double, or even treble their salary, attract the really good people into politics
This is like arguing that rock stars should be paid more so that we get the best candidate. Politics is rock and roll for ugly people. There are thousands of people up and down the country elbowing each other to get onto the councillor/party administration ladder just so they can one day have a faint hope of being an MP - a job with no educational or professional experience requirements that lets you have a decent salary, pretty good expenses, an excellent pension, and fantastic follow-on employment opportunities: directorships, lobbying, charities administration, media, industry. But more than that, you get massive ego massage and attention - foreign trips - your face in the newspaper and people listening to your every word! For the egomaniacs that make up most of parliament, money can't buy what they want.