One thing to say on the subject of trying to do affirmative action on the basis of a comprehensive assessment of socioeconomic class is that people aren’t going to agree on who counts where.
Kid number one grew up in the suburbs of Albuquerque, New Mexico. His dad was a charismatic college football star at Texas Tech but not good enough to play in the pros and eventually made millions of dollars as one of the most successful fast food franchisers in the Southwest. Mom is dad’s high school sweetheart who was a couple of years younger and stopped going to community college soon after she and dad got married.
Kid number two grew up in the suburbs of Boston. Dad’s a professor at, and graduate of, the Berklee College of Music and mom did her undegrad at Columbia and her PhD work at Harvard and now she’s on the faculty at Boston University.
Kid number one’s family is going to have a lot more money. I think kid number two’s family has more “class” in a common sense way. Which kid is inheriting more advantage from his family is a bit hard to say — it probably depends on exactly what you’re talking about in a pretty nuanced way. I don’t think this is by any means an insurmountable objection to trying for some affirmative action on the basis of a broad class metric, but I do think it’s a real stumbling block. By contrast, a straightforward attack on privilege in the form of efforts to dismantle legacy preferences and the like would have a similar effect and it’s easier to get a clear sense of what the target is. And of course if you reduce the level of inequality, you reduce the scope of inherited advantage directly.