Immigration probably makes housing more affordable, as building still a labour intensive activity, and immigration reduces labour costs. We have car assembling robots, but not house building robots, or plumber robots.

But we can't count on migration on the long term anyway, as countries with current net positive population growth rates are on the way to go sub-replacement in 1-2 generations as well.

Housing unaffordability should be pinned on NIMBY, bias against high density housing, and the decades long push to optimize housing as a rent-seeking friendly investment vehicle instead of simple dwellings. Today if you intend to buy an apartment in a high cost city, odds are that you must not only outbid people wanting to live there as well, but also rich people overseas that invest in real state as a way to stash money offshore and/or evade taxes.