I admire the way chess in Russia is still considered a pasttime for the working classes, just as bowling is considered in the US.
It's true though, that to get really good at chess, you need to have quite a lot of time on your hands, and money to fund your tournament play, study materials etc. Such a lifestyle is usually confined to the middle classes or above. Look at the great chess masters of the classical and early modern era, none of them were poor. (The Soviet masters, of course, received state funding to help them realize their goals).
If Western countries provided more funding for chess, especially for the players (and you could make a decent living off it), and popularized it more, and the image of chess as something geeky was changed, then you might see more players from the working classes become top players.
funny thing...batgirl posted something about Petrosian's youth in her blog. Take a look...think you can stack up to his "minor breeding" and "working class parentage"?
According to Nigel Short, when he defeated Petrosian in a simul at the age of thirteen, the latter resigned "by pulling a face of utter disgust, angrily shoving his pieces into the middle of the board and storming off without shaking hands." As a result, Nigel has "never succeeded in thinking of Petrosian as anything other than an exceptionally rude, ignorant, Caucasian peasant."
Some people are put off by Nigel's exceptional rudeness, but please bear in mind that his hauteur is fully justified: his maternal grandfather was thought to boast Welsh blood. You have to respect that.