I don't mind children in pubs and restaurants, so long as they are well behaved, but I think it depends on the pub/restaurant, and its purpose.
I was reading on MN about Wetherspoons having a policy of not serving more than 3 drinks to an adult in charge of a child, and that sums up the problem, really. Children running about and annoying other customers while their parents get drunk are a menace, and pubs are not the place for barely supervised children. Old-fashioned pubs where there is swearing and drunkenness are not good either, although I don't think I've been in one of those since I was a student, so they might not exist any more!
Restaurants like Frankie and Benny's are designed for children, so it would be churlish to complain about them in there; but I wouldn't be happy to pay Ivy prices to sit next to a crying baby or whining children, and with the best will in the world, they can all have their off days.
We took ours to appropriate places, on the strict understanding that they had one warning and then would be taken out if they misbehaved, and I can't remember ever having to do that (although we would have done if we'd had to). Also, 'children' can be anything from three to 15 or so, and there is a huge difference - it can't be the same rule to cover such a range of ages and boredom thresholds.
Restaurants were thinner on the ground in my childhood, but I remember my mum taking us for a set lunch (called a 'Businessman's Lunch'
) in a city centre Chinese Restaurant every now and then as a real treat. It was always orange juice or sweetcorn soup as a starter and banana or (tinned) pineapple fritter for pudding, with a choice of main from about three options, one of which was always omelette, but I thought it was so sophisticated. I learned to use chopsticks there.