There is no fixed demand for jobs, nor fixed supply of housing. Immigrant consumption creates a lot of jobs and immigrant labor creates a lot of housing
The timescale that the "lump of labour fallacy" operates on, as in the aggregate effects on employement, doesn't necessarily work for most people (individually).
Therefore it isn't really a good metric at the scale required to alleviate the problems people are facing.
"Eventually it will work out." Isn't proffering a solution.
The aggregate effects on employment caused by immigration work on an identical timescale. If your point is, "people are terrible at blaming individual outcomes on aggregate statistical phenomena", then I agree with you.