It can be the case that immigration tends to buoy wages over all, while there do exist some for whom immigration will depress wages. Again, we’re definitely trying to do this when we craft targeted policies aimed at bringing in or discouraging immigrants for specific professions, and it does have the effect one would expect.

We have a history of doing the Neoliberal “well this will make line go up and we can just help the few whom it harms” and then not helping those few, so I get why people worried their wages might be some of the ones affected aren’t thrilled. Whether most of the folks so-concerned would actually see such a thing, is another matter (I’m guessing not, in at least 95% of cases of people with those concerns).