School where my daughter teaches is 100% free lunch and breakfast, she says they throw a lot of food away because the kids won't eat foods provided, applesauce comes to mind. Kids must not be that hungry. Look around at preteen and teenage obesity, lot more kids being overfed than other wise. Where kids are starving, it is much more likely neglect than poverty; poor parents get hungry enough to find food by any means- even work, neglectful parents are probably neither hungry nor poor by world standards simply because we in the USA have been giving them and their antecedents free lunches for several generations. They are conditioned and trained to shirk responsibility.
But, the only way to catch those few that are needy is to use blanket coverage, other wise the charity/wefare will end up in the hands of the opportunists.