If you look here, you’ll see that all the trades involved in housing construction are on the list for fast-track immigration already.
As for training, we may find that it’s more the number of people leaving the trades that is the problem. It’s not that the pay is bad, exactly, but it’s an industry extremely prone to boom/bust cycles. People leave for jobs with some sense of stability. Increasing unionization and enhancing EI might be more cost effective than funding more training.