And it eliminates the stigma of only the poor kids getting free lunches.

The stigma is the point.

Conservatives believe receiving charity should be shameful.

Because conservatives (and neoliberals) think poverty is a personal moral failure - if you're poor, it's not because society and capitalism and racism and structural inequality screwed you over, it's because you, personally, were lazy or wasted your money or broke the law or didn't work hard enough.

So if a child can't afford a school lunch, it's because their parents are bad people. And shaming that child with an obvious "free lunch" (I remember having a bright red card that I had to show in the cafeteria, and the lunch lady would sneer at me and loudly proclaim "here is your FREE LUNCH" and hand me a cheese sandwich and an apple when the other kids were getting pizza just to make sure everybody knew my parents were poor) teaches the child to be ashamed of their parents and be ashamed of their poverty so they'll work harder to avoid poverty as adults.

And if schools give every child free lunch, not only do they lose that "teaching opportunity", they teach children that food is a right and that everybody, no matter their economic status, should have enough to eat, which is a direct attack on the fundamental principles of capitalism and American society.

Go into a conservative space and tell them people have a right to food and shelter and medical care and watch them froth in rage.