We all agree the restaurant has the right to do it, but we can still comment on the motivations behind it.
I think people need to be more tolerant of each other. I've been in situations with screaming children and yes it's taxing on the nerves, but children are part of our society and our communities. Shutting them out just because they are children does not make sense to me.
This restaurant's policy would make more sense if it targeted bad parenting, i.e. if a child is screaming and misbehaving, and the parent is doing nothing about it, then they should be asked to leave. I can only think of a handful of instances where I've been in populated, closed space (like a bus, restaurant, etc.) where a child was going crazy and the parent did nothing. MOST parents get stressed out by the idea that their child is annoying others and they want to put a stop to it right away.
Target the bad parents, not all children. Anyway, from the looks of that diner, they don't exactly have the look and level of service that would demand more etiquette. They're a BBQ house and the owner is a little hoity toity.