There should be immigration, but there should also be a procedure of strong assimilation, encouraging and nudging them toward becoming American American. Your remark about people looking at past immigration through rose-colored glasses is spot on. They should have been encouraged even more than they were (which was more than today) to have shed their old culture (for the second generation, to shed a culture which isn't even theirs), and immigrants today should also undergo a rigorous encouragement to assimilate. All Americans should kindly pressure them to do so, and not pander to them.
"The last bastion of the original settlers (New England, excepting the Boston area of Massachusetts) is one of the most liberal areas of the country."
When my family helped found much of New England starting with Providence, Rhode Island and Rehoboth, Massachusetts, almost 400 years ago they were already considered liberal, and have remained so since, (the region not the family).