Do you refute that importing mass amount of people into a city, without substantially increasing supply of housing, increases the price of housing?
Immigration can heavily increase demand, and so it can play a big part, depending on the immigration numbers. Anyone moving in needs a place to live as well.
Housing is also one of the few issues that is so local and immigration is such a tiny story around it to begin with. Prices are high in plenty of areas seeing little immigration activity
You’re also failing to draw a causal link here. Not to mention NYC is one of the biggest cities in the world period (10th). It’s hardly representative of most US cities.