To add on to this, the period of immigration often referenced in these arguments went from about 1880 to about 1920, so about 40 years total. After 1920 immigration was cut down pretty significantly because the country needed a long time to assimilate all those (primarily European) immigrants into the (then-primarily European) country. This is somewhat less than the 50+ years of sustained mass immigration we've had since 1965. The scale of immigration has also been much greater than it was from 1880-1920 in addition to being more sustained. The origin of immigration back then was, as I mentioned, primarily from Europe -- and like it or not, having a similar cultural and even racial background to the American population of the time made assimilation a much smoother process than it might otherwise have been.
There are a number of other things different now than back then too. For one, ease of travel and communication with one's home country and home culture has made the task of assimilation somewhat more difficult. There's the fact that before the immigration was pretty spread out between a number of different places, whereas now Latin America, and Mexico specifically, greatly outpace any other single country. There are also far fewer active efforts on the part of the United States these days to get people to assimilate.
People need to stop pretending that immigration now is just like it was a century ago. The circumstances are considerably different, and to pretend otherwise is to ignore reality.