And the demand is not just for doctors and other highly credentialed labor. The most nativist work on the Mariel Boatlift (3) (by Borjas) only found a decrease in wages for native-born high-school dropouts, and the general consensus of other economists seems to be that his work was faulty. The general consensus on Mariel (cite fn2 again) is that it was good, economically, for the native-born people in the Miami labor force when an additional 7% of the labor force suddenly arrived as immigrants over the course of a few months.
1: How much housing can be built legally is the key factor here. Immigrants can drive up the prices if no more housing can be built but that's just another way of saying that "no more housing can be built" is a really terrible policy that is enormously destructive. If building housing is relatively easy, then it doesn't drive up prices. In fact, immigrant labor is often used to build this new housing. Don't let your area become like the Bay Area and you'll be fine.
2: Wearing my physics hat, I am not comfortable saying that anything in economics is actually proven.
Some economists theorize that a long time Mayor of Boston (James Curley) used economic populism (taxing wealthy people of English descent and redistributing to poorer, usually of Irish descent) and anti-British rhetoric to reshape the electorate in ways that benefited him, even if it didn't benefit the city as a whole. The theory goes, he had a plan to drive out the wealthy people of English descent so the poor people of Irish descent would make up a larger share of the electorate and he could win more elections.
My understanding is that even most economists think that this isn't actually real. It's largely a couple of economists building a mathematical model, and then looking at cherry-picked examples, e.g., minority mayors during the 1970's and assuming that it was the result of a dastardly plan by the minority mayors rather than the result of larger social forces driving White Flight from cities.
In general, I think that mayors might make dumb decisions, but they largely do it because they think it will be good for the city and are wrong, not because they are twirling their mustache and cackling away.