The reality is progressive policies lead to better societal outcomes, one of which is housing being more efficient (either cheaper for the same quality, or higher quality for the same price).
So if your politicians haven't enacted policies to accomplish those things then they aren't progressives no matter what they say or what the media is telling you. Immigrants are not the reason Canada has the issues it has. Everywhere in the world is claiming that same issue, it's just not accurate. If governments wanted to solve the housing issue they could and swiftly.
I know Canada hasn't banned landlords and I have no doubt they're more interested in protecting the investments of the wealthy than they are interested in delivering better life outcomes for it's people. So they by definition the people in charge can't be progressives. This isn't a purity test it's a fundamental definition check. Maybe they're progressives in terms of social rights but that's only one of the government's role.