It may be a bit controversial, but I believe the male tectonic shifts going in right now under our feet are also driving much of the shift regarding “immigration” because billionaires don’t need immigrants as much anymore and they’re also causing systemic problems in things like the housing market. In fact from what I’m hearing, there is concern that immigration is now becoming a potential cost due to the major changes in society rather than a profit center by scaling up marginal returns from every immigrant you can import by hook or crook, i.e., the more people you import, the more you can undermine salaries and increase revenues and thus, profits.
The rest of your post sounded rational. This part doesn’t.
Failure to build more units (NIMBY) and properties being held as investments (non-US nationals are significant in some markets) are the two causes I would point to. Of course every additional family that needs housing drives up the cost a little, but I really don’t think that can be responsible for the massive increases in housing valuation.
The other big issue is density is expensive; it costs much more to build an apartment at the 10th floor than a detached house. Combine that with density being inherently a bit shit, you get less quality and less housing for the money spent.