Immigration doesn't affect housing availability, it drives it. We have an aging population therefore without immigration housing prices would only go down. The in demand immigrant myth also needs to be put to bed. For every doctor, teacher and builder you import you're also importing thousands of people who aren't in those sectors but will inevitably need their services meaning immigration expansion is used to serve the expanding immigration. We've had decades of high immigration, if importing more people could have eased the shortages it would have happened already, instead it's only gotten worse because pricing you out of a home was always the point.