It is self evidently true that the demand for housing migration creates increases prices and reduces availability for housing. Regardless what you think about migration policies, denying such an obviously true statement makes you look ridiculous.
What? They live in the same type of building I live in. Their rent is paid by the government if they can't pay themselves. Have you never been to a major German city? Visit any part of the city which isn't for millionaires only and you will find it inhabited by locals and migrants. You are completely delusional if you think otherwise.
>and the city economy would shrink too and then you'd complain about that...
This is irrelevant to the economics of housing. Migrants drive up housing costs, that is the most basic economics.