More importantly, how exactly is an undocumented immigrant paying taxes? Do they somehow have a tax ID or SSN without documentation? Or are you just referring to things like sales and fuel taxes?
So the immigrants are paying into the SS/medicare system, but they don't receive any of the benefits.
I think the solution to the housing problem is to increase supply. Not to try to prevent a whole group of people from being able to buy homes.
That seems totally reasonable if there's demand for more houses and the ability to build more.
Phrasing it as though people who already can't to buy a home are being prevented simply by not providing government assistance is a big disingenuous though. Preventing them implies that they could otherwise buy the home, and if that were the case a government program wouldn't be needed.
Also have you talked to any Gen Z or millennial? Most cannot afford housing, why in the world would you introduce even more demand for scarce housing?
And supporting migrants isn't introducing new demand, it's just exposing the demand that's there. Like, the migrants are there and they need housing.
The us vs them attitude is an artificial division. We can, and should build housing for all.