recently its because we need skilled and cheap labour to fill the holes in our health and buisness system.
Housing shortages is not a direct immigration issue[1], its a policy issue. People of a certain type do not want high rise buildings near them. people of another different type are really happy because they can now build less and sell for more. Yet another class of people go apoplectic when its suggested that social housing should be built (my taxes! why do they get something for nothing.)
In short, we are our own worst enemies.
[1] net migration at 1 million was utterly insane, and I'm still not sure how a government that lead brexit with "we'll control our borders" managed to be so divorced from what it was doing. Thats rhetorical by the way its short-termism and utter utter incompetence.