On a related note, I suspect cities primarily relying on cost-of-living arguments to attract people may actually be subtly signalling the reverse of what ambitious people are drawn to. A city's primary signal should be the vibrancy of the place, not the low cost of living.
In my old rust belt city, many were proud of the fact that folks who left for New York City were coming back because we offered a low cost of living. But the people returning were also those who had "given up" on their ambitions... whose dreams were broken by NYC. I realize this is a bit of an unfair generalization, because it's not universally true (some ambitious chefs who returned -- and they are very good -- were those who trained in NYC but simply found NY's restaurant economics too oppressive). But supposing we were subtly attracting folks who had "given up"... in large enough numbers, they are likely to set a certain cultural tone for the city. Great cities are not built on low cost-of-living is all I'm saying.
Side: as a historical example, consider Canada vs U.S. -- Canadians are generally a more risk-averse people than Americans, and it's partly because the United Empire Loyalists were American loyalists who chose the safer route of siding with the British, whereas U.S. Americans forged their own path. The Loyalists set the tone for much of the country.
As to your second point, cities with ambition tend to be expensive because people want to live in them, often in spite of the expense. However, the reverse is not true -- just because a city is expensive doesn't mean it attracts ambition. Nantucket's pretty expensive but it's not brimming with ambition. Not sure about expensive cities correlating with class... New York is very expensive, but still manages to attract ambitious working class people who are trying to make it there.