You're partly correct, but (possibly?) incorrect because of the "socio" part of socioeconomics.
You're probably right that it's relatively rare that a family couldn't save every last penny possible to pay for at least a little of those benefits (though still not nearly as much as other families). Although there are families where parents are already working crazy hours and still really don't have money to spare, where a surprise $300 car repair means rent will be late and the next few weeks of food will be peanut butter sandwiches and spaghetti.
But there are two factors on the "socio" side that also hinder things even when finances are a little less extreme:
1) Families that lack a strong educational tradition may not really understand the dynamics of what is necessary to achieve the highest levels of academic success. Sort of like not having strong financial planning skills because they were never taught to you... You don't know what you don't know, so you aren't aware of what it takes to be successful.
2) There's at least a little bit of a mental health aspect to things. I can't easily put it into words, (and I might be wrong) but it seems like it could be bad in some ways to exist on that extreme of a level, for kids to practically never see their parents because they're always working, and for the family as a whole to have little ability to catch their breath and have... I don't know, a bit of decompression. Again it's hard to put into words, but I can say that I grew up pretty poor, and it was stressful as hell even without being at the very extreme edge of things, needing food banks etc.