This is an interesting strategy - essentially it would undercut the current system that depends on the full-price kids to make the finances work (a driver of the food choices: they are designed to appeal to the kids who pay full-price in order to be financially sustainable. Since parents don't typically make the choices at school, sugary, salty, fatty foods "hook" more full-price kids.)

The summer lunch program works on this model, though it is localized to areas that qualify - and the food is better. The breakfast program works on a similar model, except that it isn't a very popular program - so again, the food is driven by the kids' taste preferences and not by good nutrition or quality. (More kids = more reimbursement = more possible economies of scale)

A simpler policy solution than universal lunch might be to equalize the payment/reimbursement amounts - right now, the federal government reimburses at three different rates, and there are regulations that determine how much can be charged beyond the reimbursement rate.