Overstaying a visa is unauthorized. Paperwork mistakes are not a majority. And when people say "undocumented immigrants", almost every time they aren't talking about people who came in from the U.K. and overstay their visa (which they got because they were staying for longer than 90 days)--but again, once the term of the visa is up, they're not undocumented (indeed, the visa is documentation), but rather at that point they become unauthorized. Because of the visa, it is documented that they are unauthorized.

And they're almost never talking about someone who accidentally didn't get a stamp in their passport when authorized to enter.

I would agree with you if the points were accurate, but they're really not.