Moreover, if free food is only available to low-income students, having to eat that food can become a symbol of poverty, and some students may feel ashamed to receive it. Making it available to all students, without reservation, avoids that.
People say that, but it’s not like they have a flashing light that says “TIMMY IS POOR AND THIS LUNCH IS FREE,” they just punch their code or swipe their card or whatever and it rings up as free.