Children do not learn without practice! When my son was young we would "put our manners on at the table" and he learned proper etiquitte for eating anywhere. I was frequently complimented on his manners when we were in public. Sometimes it helps to make a game of it but you do your children no favors if you fail to teach them good table behavior!
My parents had no education in that regard - I went to a parochial school and the nuns taught me good table manners. The one thing they didn't teach me was how to eat chicken with a knife and fork! It was always finger food at home! My mother never served chicken that wasn't fried.
The first time I had dinner at my InLaw's house of course they served chicken...what a nightmare! After that hubby taught me and my son & I learned together how to finesse the meat off a chicken breast with no fingers! For several months we had practice chicken at home.
Good manners is only the art of making those around you comfortable.
When we were out in public I would take my son into the ladies room to discipline him - (at a very young age, of course) and in a short time all I had to say to remind him to behave was "do you need to go to the restroom?" Instant compliance! To this day he claims it was because his food would have dissappeared when he returned!
As an adult I took a course in table manners - until then I considered myself well mannered - there are always things to learn!
When I go out for a nice meal I have zero tolerance for screaming babies or rude children. Take them home and teach them how to behave! DH accuses me of giving "killer" looks to the parents - guilty! He's lucky I stay at our table!