I know I paid a nominal fee for my daughters school lunch program.  They never mentioned who did or did not pay, I never heard of shaming related to lunch programs.   +plenty of other shaming of course, as this is when teenagers decide they need a pecking order.

A case could be made that parents who can afford paying for their child's breakfast will continue to feed them at home.

I absolutely believe no child should start the day hungry.  I just don't buy into the shaming arguement. 
 
There are a hundred things a child can be mocked for - as we both know.  I was tall and fat, no place to hide..😣What the child needs is family that loves them, which cushions these outside taunts.   *Easy for me to say, very hard to provide.   We aren't born with life insurance coverage for rotten patents.    

*Bad parenting can happen at all incomes.  Poverty multiplies the stress on the family though