Perhaps, but the Founding Fathers were not silent on the subject of immigration. Article 1, Section 8, Para 4 states: "Congress shall have the power...'To establish a uniform rule of Naturalization,..". The issue of who were able to become citizens was not left to being a free for all, as you imply.
Furthermore, under these powers Congress passed a number of immigration and naturalization laws, among those being:
A. The establishment of "Special Ports of Entry for Immigrants", notably Ellis Island from 1892-1924, its peak years until 1954. Even if an immigrant reached Ellis Island, they could, legally, be turned back to their country of origin.
B. The Johnson-Reed Act of 1924, also known as the Permanent National Origins Quota Act, began a quota system that lasted for 40 years.
C. A preference system was added in 1952 with the Immigration and Nationality Act, giving preference to people with skills needed in the United States, relatives of U.S. citizens, etc.(the relatives of German rocket scientists).
D. The Hart-Celler Act or the Immigration Reform Act of 1965 did away with the quota system, though it kept much of the preference system. (the breach in the dam that led to its breaking).
You are implying that the Founding Fathers were silent on immigration and the fact of its consideration proves otherwise. They were concerned who would become citizens and gave Congress the power to decide who could stay, who could become a citizen and who should, or could, be deported.
There is a boat leaving for France arriving at LaHavre. Au revoir. Bon voyage.
1892 is a long, long way from 1787.