For me, there are two aspects to this problem: crying, and improper behavior in public.
Babies, and other individuals under the age of 2/3, shouldn't be allowed in diners, restaurants, theaters, and some other public places as a general rule, because of their incessant crying. Children of 2-8 years old are allowed, but if they start crying, out they go.
Then, for kids of the age of 8 and under, it goes under "improper behavior in public". What constitutes as "improper behavior", for me, is the same than "annoyance" for our friend, OP: when your kid starts annoying others than your own family or friends with whom you're sitting at a diner/restaurant. Same thing: three chances, then out he goes.
On public transport, it's a more difficult situation: you can't force a kid off a plane, can you? For these cases, it's the staff that is going to deal with kids: isolate him, switch him places, and just make sure he can't bother anyone anymore. And, as a general rule, forbid 2/3 year old children from boarding a plane or train on first or business class: I'm paying extra to have extra comfort, so I don't have to put up with a kid crying all the time.
To sum it up, 2/3 year olds and forbidden from going to the movies, diners, restaurants, and first or business class on planes. 2-8 year olds have three chances, then out they go if they start being annoying/crying, and get isolated if you can't actually take them outside (e.g. on planes or trains).
Babies, and other individuals under the age of 2/3, shouldn't be allowed in diners, restaurants, theaters, and some other public places as a general rule, because of their incessant crying. Children of 2-8 years old are allowed, but if they start crying, out they go.
Then, for kids of the age of 8 and under, it goes under "improper behavior in public". What constitutes as "improper behavior", for me, is the same than "annoyance" for our friend, OP: when your kid starts annoying others than your own family or friends with whom you're sitting at a diner/restaurant. Same thing: three chances, then out he goes.
On public transport, it's a more difficult situation: you can't force a kid off a plane, can you? For these cases, it's the staff that is going to deal with kids: isolate him, switch him places, and just make sure he can't bother anyone anymore. And, as a general rule, forbid 2/3 year old children from boarding a plane or train on first or business class: I'm paying extra to have extra comfort, so I don't have to put up with a kid crying all the time.
To sum it up, 2/3 year olds and forbidden from going to the movies, diners, restaurants, and first or business class on planes. 2-8 year olds have three chances, then out they go if they start being annoying/crying, and get isolated if you can't actually take them outside (e.g. on planes or trains).