So you're right, it is totally, 100% unrelated. Otherwise you're making a claim that one of the poorest demographics in the US with the least amount of home ownership is somehow responsible for a massive price spike in housing over the last few years.
Also, most of the remaining 11.3 million undocumented still occupy housing. So out of 340 million Americans 35% don't own property leaving around 119 million non-home owners. With 8.3 million undocumented not owning homes per your numbers, undocumented are adding approx. 7% to the needed rental market. Again, what does that increased demand do to housing pricing in a relatively inelastic constrained market? I'm personally for more open borders but I'm also for being honest about what that means including increased housing stocks.
If there are 10 houses and 12 people. The houses will go to the 10 highest paying people and so the 2 poorest immigrants will be left in the rain.
The problem is not that we now have 10 houses and 15 people (3 from immigration) it's that there's 8 houses now [1].
[1]: https://www.fanniemae.com/research-and-insights/perspectives...
Especially after 2008 almost all house valuations in the US are based on square footage and number of bathrooms. So bigger houses and more plumbing is the only way to make money as a developer.
The latest available US census data for rentals is here, and in many use cases rentals drive home sale prices more than the reverse: https://www.dhs.gov/sites/default/files/publications/immigra...
No one made a claim about federally guaranteed home loans except you, and you can’t transfer a claim from home loans in general to federally guaranteed home loans, because the latter are only about half of the former. And actually the claim wasn’t even about home loans to start with, but about home ownership.