Really silly argument.
The Founders may not have supported unfettered immigration in principle, but somehow this country didn't get around to putting ANY restrictions on immigration until 1875. With significant restrictions not being passed till the 1920s.
IOW, meaningful restrictions on immigration were in place for only about 50 or 60 years out of our 240 year history.
Such restrictions have excellent arguments for their passage. A consistent American tradition of restriction is not one of those arguments.
The original United States Naturalization Law of March 26, 1790 (1 Stat. 103) provided the first rules to be followed by the United States in the granting of national citizenship. This law limited naturalization to immigrants who were "free white persons" of "good moral character". It thus left out indentured servants, slaves, free blacks, and later Asians. While women were included in the act, the right of citizenship did "not descend to persons whose fathers have never been resident in the United States...." Citizenship was inherited exclusively through the father. This was the only statute that ever purported to grant the status of natural born citizen.[1][2]
The perception of the need for newcomers to settle the interior of the country was completely different in the 20th century than it was in the 19th-and especially the last part of the 18th century during the inception of the United States.