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Sure, I'll give you another piece of the puzzle: reading (and language deficits in general) start young. Like, very young-- by age 2, you'll see a difference in working class families and upper class families by "6 months of development" or more, depending on the study. (I lost my Zotero citations, but you can search Google Scholar for "differences in vocabulary by socioeconomic status of toddlers" to find a few).

Experts try to offset that by promoting not just preschool, but early childhood education from birth onwards. Of course, widespread implementation stalled in Congress but you can still see some districts with at least free education at age 3, and you do have (or had?) language support for toddlers through disability services. It's very minor compared to the need, though.

That said, there's still a billion other factors. Free breakfast and lunch at school, for instance-- easy enough to pass in a sane state, makes a tremendous difference at all grades. Parent involvement programs that are sensitive to parental schedules (like night shifts) and home language and so on. It only makes a dent, though- a statistically significant dent, but until family life isn't as stressful and difficult for the working class, it's a bandaid over a gushing wound.