The biggest drawback to first class is that you're much less likely to get nice seatmates.
Not to mention: This isn't a ban on children, merely children under 6.
It is perfectly legal and acceptable to throw someone out if they or their party are being disruptive.
Banning them simply on some artificial criteria is a bad idea.
You would have to actually enforce the rule, and that would be ugly, but it would have been way, way less ugly than what's happening right now. And it would have been just as good for publicity, if you advertised it.
I see your comparison more in a quite different light which I often encounter; if you let kids in, why not dogs? My dogs don't bark, whine, poop, pee on the floor, don't run around, don't break things, don't bite things, don't beg for food (or icecream) really loudly; something which I have seen young children all do (yep including the barking, seriously) in public places, including airlines and restaurants.