> No, immigrants do not have abysmal rates of employment

I think it depends on the specific form of migration to that country. For example the USA or Switzerland have very high skilled people coming to their country. In Germany too, but there was and is also a large influx of very uneducated migrants. Currently 55% of all welfare recipients in Germany are people with a migration background which shows that the opportunities/capabilities are not equal.

I do think though that welfare can still work with free movement, it is just that there would need to be a similar level in all/many countries.