You can’t be an “illegal” or “undocumented” immigrant unless it’s possible to be an immigrant even without documentation or following a particular legal process. The reason you want to stick to arcane language is it facilitates dehumanizing these people. The word “alien” indicates there’s something significantly different or scary about the person being discussed. There isn’t. No one uses the word “alien” to describe people from other countries today, except in this instance. But no matter how the Constitution uses the word, until the 1880s the US had virtually no immigration laws.