I don't think most progressives assume stereotypes are arbitrary or random. The standard progressive view seems to be that the stereotypes were based on the fundamental attribution error: attributing negative traits that a group has to their innate nature rather than to negative influences.

Example: Members of group A enter region dominated by group B. The educational system in region B focuses on entirely different languages and historical periods than the educational system in region A. Group B considers intelligence to be demonstrated by a mastery of language and history. Group B then assumes that members of Group A are inherently unintelligent, instead of assuming that there's a good reason why otherwise competent members of Group A are completely inept at Group B language and history.

Group C denies group D access to certain types of training for reasons that are, at the time, valid. The reasons become invalid. Group D asks for access to those types of training. Group C points out that Group D currently has no demonstrated skill at those types of tasks. Group D is being stereotyped as inherently bad at something, when in fact they are merely untrained.

Why don't progressives put the same effort into dismantling insulting stereotypes of their political opponents (certain types of rich people) in the same way that we do for other groups? Presumably, because rich people can afford to hire PR agents, and are not suffering harm from stereotypes*, and because people generally do not make their political opponents look better.

*I haven't seen evidence for rich people being harmed by stereotype threat, but if some study shows they are, please link me to one.