Pigeons only account for one of those 200+ species the OP mentioned though (unless they live in one of those regions with the native Band-Tailed Pigeons). That particular kind of pigeon that everyone loathes (European Rock Pigeons) aren't even native to the New World and are an evasive species.
Ironically, they're in the decline in their natural habitat while thriving in North/South America. The Americas has a few native pigeons, but none are as well known or wide spread as the Rock Pigeon. Quite a few pigeon species are also far from the ugly "rats with wing" stereotype[1], but they don't live in Europe or North America.