The Immigration Reform Act of 1965 changed preferences from the well-educated and higher-civilized from countries whose populations happened to be advanced and pale of skin for just the opposite. I believe Evil Ted and his cohorts (under the protective umbrella of Johnson’s “Great Society”) took advantage of the chance to alter the constituency.
The fact is, the eastern Europeans are going to places like the UK and France, which are VERY CHEAP to fly to when compared to the U.S. Southern Europeans aren't reproducing, although it seems to me that every wealthy Italian keeps a place in Manhattan or Miami these days (the poor Italians being kept fat by Italy's generous welfare state).
Even if we abolished all visa/residency requirements for Europeans, we wouldn't have a much larger influx than we would now. If we gave preference to the skilled and educated, we would probably see more immigration from Asia, with a very slight uptick from Europe.