I think the argument is right, but the evidence you cite isn't very supportive.
> Most of the wealthiest countries sustain high immigration ... Switzerland has over 3 times the US rate of immigration yet has one of the highest bottom quintile incomes in world.
Obviously economic migrants aren't going to migrate somewhere with no opportunities. It is hypothetically possible that low-skilled they choose the country to migrate to by picking the one with the highest bottom quintile income (some proxy of, given they wouldn't actually review the stats). If the host country has immigration limits the economic state of the hosting country would change slowly so the migration wouldn't immediately normalise the host countries economy with the rest of the world.
Basically, we'd expect to see that statistic even if migrants did lower the income of their quintile. They don't, but the evidence of that is a separate thing.
> Most of the wealthiest countries sustain high immigration ... Switzerland has over 3 times the US rate of immigration yet has one of the highest bottom quintile incomes in world.
Obviously economic migrants aren't going to migrate somewhere with no opportunities. It is hypothetically possible that low-skilled they choose the country to migrate to by picking the one with the highest bottom quintile income (some proxy of, given they wouldn't actually review the stats). If the host country has immigration limits the economic state of the hosting country would change slowly so the migration wouldn't immediately normalise the host countries economy with the rest of the world.
Basically, we'd expect to see that statistic even if migrants did lower the income of their quintile. They don't, but the evidence of that is a separate thing.