Sure, but there's a difference between a fear being tenuously plausible but overblown/stoked, and a fear being flatly incorrect and inverted. The former is harder to counter. The latter still has the standard problem of lies propagating faster than the truth, but it at least theoretically should be countered by the truth.

At least in theory, many people perpetuating this myth are at least nominally convinced that it must be obviously true, because the zero-sum thinking is simpler than the reality.

Of course, the other problem is people going "my anecdote beats your study", because they personally know someone who blames immigration for losing their job, and a study of people in general still lets someone go "ah, but that's not always true".