The recent slump in sales would prove your point wrong. Home ownership is at its highest levels because of record low interest rates, not because of your illegal buddies.
You know, you're right! I'll instead thank international trade for the low rates. Foreigners send us goods and their capital (by investing in our debt instruments) and this keeps the rates low. Thank you for correcting me.
I do wonder, though, just how much new home prices would be, how many of them would get built, and what this hypothetical new scarcity of homes would do for the affordability of homes if immigrants weren't able to work on the crews that built them. Do you suppose the cost might just go up?