I have a different perspective. In my experience, even with very little income, or even none, children can still learn. If they do not have enough food, that is a different issue, and one that isn't mitigated by public schooling anyway. I'm an autodidact, and my children are mostly that way as well, since dp is too, and we do not hinder our children from exploring their interests. In any case, I guess I'm not sure what sort of scenario you're referring to wherein the children would have access to absolutely nothing. If the situation is that a family cannot pay rent without sending their dc to gov't funded schools, then, imo, an examination of values is definitely in order, with quick and diligent action to follow.
I don't share this from any ivory tower! I consider the children being with their parents as the default position, so at base, this is not only possible, but preferable as long as there are not competing values that are yet undecided in hierarchy. There's just no way my family could have stayed where my first three children were born, because we just couldn't sustain an income commensurate with the cost of living and jobs there. It wasn't a matter of whether we could if we sent our dc away, because our values were in order that demonstrated that we had no desire to contravene the natural law. Certainly it was well within our right to do so, if we wanted to, but we didn't.
So, for us, the choices were never between sending our dc away so we could have more income, or keeping them home and starving! The choice was: where is the best location to achieve a family life the way we want it? This necessitated finding a location with a lower cost of living and jobs that paid enough on one income to meet the bills, so there also had to be much less competition for those jobs.
It took two moves to end up where we are now, but we moved the second time for the same reason. Our income is presently triple what it was when we had three babies in the city, our cost of living is now the same as then, but we have a seven-person family, so percentage-wise, it's much less than what it would be there, now. Even still, we are in the process of unjobbing in order to further align our life with our values. So not only will our dc be home, but so will we. The amount of income we need and/or want dictates how much and what types of work we'll do to earn it. Our dc being home is just a given. The two issues are not in conflict. At all.
We have friends from our old city who have given up their plan to hs because of finances and the recession there (which never hit here, and won't- another reason we moved here; it's in a bubble); their actions show that they value paying for their house and thereby keeping it as a higher priority than homeschooling or having their children close with them on a daily basis. Fine. But it is not the only option; it's the one they choose, mostly out of fear, but obviously it could have been chosen under better circumstances as well. But it is a choice, not a necessity or a requirement of some unchosen greater ideal than the ones they hold.
To me, income generation and living my daily life alongside my children just cannot come into conflict. It just isn't possible because my hierarchy of values is firm, well-examined, and informs how I think and thereby, live.
I'm looking forward to a further explanation of this:
I'm very surprised that homeschooling families would risk their child's long term education by homeschooling no matter how dire their finances.
I do not have any partnership with the gov't or its agents in the education or raising of my dc. There will never be a time when I weigh my decisions about my family by considering whether or not gov't programs and hirelings could do a better job than me. From my perspective, such a thought process is completely antithetical to being a mother, an adult, and an autonomous being. Obviously not everyone would agree, but I'm surprised that anyone would assume that all parents of children would consider themselves beholden to even consider worth consideration what the gov't "offers" their children.
It wouldn't matter to me if they offered space camp one week, an archeological dig the next, a trip around the world, fifteen languages and pastured goat stew for lunch. It's completely irrelevant to me. My only concern about the gov't in this regard, is that its agents not interfere with the loving, healthy family life that I have worked so hard to provide. That's it. You suggest that I should consider whether I can do better than the school system or not, but my calculated involvement with it is to comply with regulations in order to reduce to the greatest possible extent, its influence and hindrance to my family's peace.
This would be my perspective no matter where I lived, no matter what my income, because my underlying philosophical position/thought is that the well-being and education of my dc is my own responsibility. Period. This means that if I see that my dc need someone to show them something, I seek that person out, evaluate their character and expertise, and decide if s/he will meet the need. If so, I hire him/her, or better yet, work out a mutually beneficial arrangement that allows for relationship with this person beyond the formality of employment.
A school system is as relevant to my life as fast-food and tv programming: when contact is unavoidable by unforeseen circumstance, I make my best attempt to disengage quickly and politely, then mitigate the damage, if any, with care and tact, and resume life as usual (for us).