I personally grew up in a single minimum wage income household, which supported 2 adults and 3 children. I'm very familiar with dubious food. That said, it is quite reasonable to meet the caloric needs of a grade school-age child in the United States with little regard for taste or variety for roughly a dollar a day. Adding some variety and nutrition to that wouldn't be much more. Formula for very small children averages between about $40-$100 a month so they are a little more expensive, but you did specifically say school children.
There are 50 lb bags of rice at my local grocery store for $15. I have bought them before, they're short grain and not very good so you wouldn't want it everyday, but in a pinch you can meet your caloric necessities with rice for basically cents.
I actually don't have a very strong sense of taste, and manage to get by on ~$3-$6 of food a day, my diet consisting largely of over-hard eggs, canned fish and canned string beans. I am quite confident I could get by on half of that if need be.
As far as homelessness goes, I don't have reliable statistics on this, but depending on where you get your statistics between 20 and 70 percent of America's homeless have mental health issues.
I am perfectly aware that this is anecdotal, but I have a friend who worked for an organization that housed homeless in unoccupied Homes and Apartments. According to her, the majority wandered off because they did not understand what was going on. In Minnesota no less, it's cold.
My point is simply that there is a sizable portion if not majority who cannot be helped without stepping on their personal freedoms, because they won't allow it, for better or worse.