My thing is, even if its not the main issue causing issues with demand, I don't see how you can bring in nearly 3 million "undocumented migrants in a year" who are all going to compete for the lowest tier of the housing market, and outcompete single parents, etc on what they can afford in rent (since many have several adults rooming together and sharing rent). If you only have so many affordable apartments/houses etc to go around in a town/neighborhood/borough, etc, an influx of a few thousand, especially in high population areas is going to drive rents up and create constraints on the market.
https://www.nbcnews.com/politics/immigration/migrant-border-...
It's certainly not helping. I don't see how you can not link these two issues, if you bring in migrants, documented or not (or not migrating through legal methods), they need housing, and its going to be the undocumented ones competing for housing in the sector with our most vulnerable citizens.
There seems to be some level of cognitive dissonance politically, at least in the media, when it comes to these two issues. Maybe those on the ground notice it or have tried to call it out. But if I can see it from my middle class suburbia, I don't see how this is getting missed, if not outright ignored.
All that to say, while certainly a local issue, this is a national issue I feel is at play here that local governments (short of stunts like bussing ppl to Martha's Vineyard or NYC), have no control over.