The problem with the logic of that position is that it implies that the Constitution is the source of man's rights, which it is not. The Constitution doesn't give anyone any rights, it only restricts the government's powers to infringe upon those rights which already exist. The source of all men's rights is not the people, the government, or the Constitution, it is God. It is generally held that "rights" are given by God to all men not just Americans. This is the principle of Natural Law upon which this country and the Constitution are based, and this principle is openly declared in the the Declaration of Independence which says:
"We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights..."
Privileges of US citizens (e.g. voting etc.) certainly can be restricted to Americans because they are granted by law. Rights, however, cannot be restricted, because they are universally given to all men by their Creator.
It amazes me that so many folk, supposedly knowledgeable about our society, blow it by conflating rights with privileges.