I would LOVE to live in a city like that. If the city has everything I need along with things like a zoo/aquarium and a job, I'd probably never leave it. Personally, I don't like kids, babies, or most teens. I avoid them even in my own family and I have no desire to have any of my own. A city like that would be perfect for me, probably.
Also, I don't think the OP is saying that kids are wiped off the planet. They are kept outside city limits. And people can move in, so the population of the city can remain steady as long as there are people coming into it.