I spent years teaching subjects which required the development of a strategic understanding of the field. That is, to consider not only the mechanics of their job but the political and social implications and contexts of that job, (it was a politically sensitive area).
While it may have been a failure of my teaching, in end I started to think that some people struggle with big picture thinking - either their training, their experience or their wiring hinder their capacity to do strategic analysis. This is not to say they were unintelligent or uninvolved, but that this was a tool they didn't, or maybe couldn't, master.
Feminist and class analysis require the strategic analysis of systems and groups. It is difficult to consider the impact of systems if your thinking keeps bringing you back to individual cases.
I think this is why, when you say, "women are more oppressed than men (as a class), someone will always chime in with, "yes, but I know a woman who is the CEO of a corporation, so how is that oppressed. Hey? Hey?"
Class analysis depends on making broad generalisations in order to discern the systems which create particular conditions. Identity politics is the exact opposite of this kind of thinking because it privileges individuals. Not to say that personal experience isn't important, but the struggle or success of individuals only tells you about those people. Class analysis tells you about what the structural mechanisms are which create oppression.
Identity politics is a logical outcome of the myth of the individual, and it is, as was said above, a useful tool for maintaining privilege.
It's why feminism and Marxism are such powerful tools for the analysis of society: and why they face such hostility.
I don't know how to fix the thinking bit - a concentration of the development of thinking skills in school maybe?
Sorry if this is a bit incoherent - I'm typing on my phone between chores.