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i'm less sure of that than you are IanO... the climate in london is mediocre at best but it's still a pretty strong destination city.
in fact, the fastest growing cities in europe are balashikha, tyumen, tirana, oslo, sochi, coventry-bedforth and stockholm. none of them, except perhaps for tirana, are renowned for their climate/weather.
9 of the top 10 fastest growing cities in the world and17 of the top 20 are located in africa and none of them are particularly well renowned for their climate/weather either.
climate may see someone elect not to move somewhere but it's pretty seldom - unless you're retired or independently wealthy - that climate is going to be the primary reason for moving anywhere.
i think that's why some of our "winter city" efforts are misguided when they are directed elsewhere and not at those who live here. no-one visits, never mind moves here, to enjoy winter per se. they will visit and accept moving here when there are programmed events and activities directed at enhancing the lives of those who live here that become worth taking part in.