He can eat where he likes. But his reasoning betrays him. This isn't like state school funding coming from general taxation on all taxpayers, or whatever he thinks. The eateries that do kids-eat-free aren't doing it as a charitable endeavour, and they are not wannabe Robin Hood taking money from the childfree guests to give food to the children!
They do kids-eat-for-free for one simple reason: they make more money that way than they do charging parents for their children's food directly.
They edge out their competitors by offering these free meals, gaining more customers, while at the same time keeping their costs low with set kids' menus. The options on the free-meals-for-kids menus are usually things they can buy in bulk, buy cheaply, and store well (avoiding wastage).
Meanwhile, the parents, feeling expansive due to the savings on the children's food, spend their savings on extra drinks (including soft drinks) for the family and by ordering dessert.
Your brother needs to stop feeling sorry for himself and pretending he's being exploited. The people funding free kids' meals are... the people who pick these places because of the free kids' meals, i.e. parents.