"Person" does not indicate one's foreign origin. I can be an undocumented "person" and yet still be a citizen... I just don't have documents.
So "person" is imprecise and not a meaningful source substitute for "alien" or "foreigner".
Words have precise meaning. It's silly to be offended by words when they are precise and not intended to offend. "Illegal alien" might be offensive if used in an unkind way, but so could "undocumented person". Our society would do well to quit assuming intended offense at every chance. It's a legal issue with a precise legal term.