You are overstating the intersection, hence the reason I used the term "social capital" because I safely presumed someone would respond with this line of argument. Further, social capital in my opinion and the way some academics use the term is not simply about income but also relates to culture, recreational habits, spending habits, and most importantly access to networks. And so, the teacher and the plumber generally speaking are not of the same class once you look beyond income.
But there are some exceptions when you look at race and geography, but not enough to undermine my general point.