Just thought another factoid might be useful to get a sense of the American tradition of immigration.
In 1921, the US incorporated the Emergency Quota Act into law, which embedded the National Origins Formula in its immigration architecture.
In 1924, it issued the Asian Exclusions Act, which operated like the label on the tin.
The National Origins Formula ran until 1965 and limited immigration from various countries/cultures.
So there's hardly a long tradition of avoiding racial/religious tests of immigrants.
No one is advocating a return to the old formulas. But sometimes you need to mix the immigration potion more carefully than simply to say "Thou shalt have no religious test," as if it is set in stone and irreduceable.
Would folks tolerate immigration by Indians who think it is fine to practice suttee or Aztecs with their tradition of human sacrifice? Of course not. So actually there is a test in play in relation to the definition of religion.