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.