Some upscale restaurants have dealt with this issue by posting a discreet sign just within the entrance: "Chez Pierre is not a 'family' restaurant. We have no high chairs, booster seats, or children's menus. If you do bring children, they must remain well behaved, or your party will be asked to leave."
I know a mother who has conscientiously taught her children to sit still, keep their voices down, decide promptly what they'd like, and order politely from the menu. They're not allowed to complain, argue, or carelessly spill their food. They thank the server when their meals are brought.
Other patrons have been known to stop by their table with compliments on how nicely the children behaved. The servers often bring them free desserts, or give them lollipops as they leave.
Airline travel is a whole different issue. It's asking a lot of even the most placid children, to force them to tolerate long security lines, long corridors to the gate, long waits to board the plane, tight seating with no permission to move, relentless whine from the engines, and few interesting things to occupy them. Many babies cry because of the pain the pressure changes cause to their ears. It's very hard to feed a small child or attend to diapering or toilet issues right away, especially since the "fasten seat belt" signs are on so much of the time. And of course the mothers' usual methods of discipline, ranging from "time outs" to yelling or spanking, aren't possible in an airplane, so they just throw up their hands and ignore behavior they can't prevent.
Airlines used to try to make child passengers feel welcome, with special meals, coloring books and crayons, and little pilot's wings or hats. Now they're just considered a pain in the neck.
My suggestion (which the airlines will ignore): Don't have "No child passengers" flights. Instead, promote one-per-day "Child-friendly" flights. On well-traveled routes, with several flights a day, a late-morning flight would have fewer business people and more leisure travelers on it anyway.
Of course, with today's hub system, most air travel involves one or more connections. But any little tweak to make air travel more comfortable for both adult passengers and children would be most welcome.