He was brought-up working class. His father's a Jamaican-born mechanic who moved to the UK in the 60s. He wasn't born with a silver spoon in his mouth. The class-war stuff probably isn't helpful.
No, the goal was to keep it tilted in capital's favor. Which group wields greater decision-making power? Which type of income is more lightly taxed? Capital vehicles (corporations) have almost all of the rights and practically none of the responsibilities - or vulnerabilities such as mortality or geographic limitation - of human laborers. Yes, the playing field needs to be leveled ... the other way.
It seems lots of people commenting here want Google to be a sinecure program for superfluous engineers. They're ardently pro-capitalist when it comes to startups and VC money and fleecing the proles with predatory advertising. But curiously left-wing and we're-all-in-this-together when their privileges are threatened.
I think painting this as a "capitalists vs the left" thing is reductive and unhelpful. Personally, speaking as someone who Americans would probably consider extreme-left, I very much agree with your second paragraph, but I don't see this as "capitalism vs the left", I see it as the elite rich abusing their position to the detriment of everyone else. A story as old as time.