I think you're confusing social class with Marx's proletariat (working class) vs bourgeoisie (capitalists).
Your bargaining chips are determined by supply and demand. If you have stuff - money, knowledge, skill, power by virtue of organizational position - you have leverage. If you don't, you don't. Eliminating social class doesn't change this much.
if you have skill and knowledge that others don't, but others need, then you can bargain for your meals and shelter very efficiently - so much so that there would be left over money.
The people with connections and serious money do not meaningfully compete against their own class (at least, they ensure that their intra-group competition does not benefit anyone outside their group). The people with skills, knowledge, and work ethic have to compete against the whole world.
Your bargaining chips are determined by supply and demand. If you have stuff - money, knowledge, skill, power by virtue of organizational position - you have leverage. If you don't, you don't. Eliminating social class doesn't change this much.