I'd recommend you take a closer look at the demographics of the rest of South America. Brazil, particularly in the country's Southeast, has a German and Italian diaspora comparable to that of Argentina. Chile received a similar degree of emigration, from all across Europe. Peru has historically been host to large Asian and European immigrant communities.

In the late 19th century Argentina began experiencing an economic boom due to huge agricultural exports (wheat/sheep wool & meat) and was a popular choice for European investment - for example about 50% of British foreign investments during that period were in Argentina. By 1913 its GDP per capita was one of the highest in the world, on a par with that of France and Germany. It was thought that it would become 'the United States of South America' in terms of economic power until WWI broke out and severed its trade links to the faraway countries upon which it replied for its export economy.

Economic immigrants aim for richer areas where they perceive they'll have greater available opportunities. So you have poorer Europeans sailing to late-19th century Argentina, or to Brazil at that same time during its Amazon rubber boom. You also see intra-country ec. migration like rural Chinese flocking to cities or Americans a few years ago heading to the shale oilfields of North Dakota.

People go where they hope the money is.