I don’t like it because it’s dishonest. The term “undocumented” implies that something has been agreed to or sanctioned, just not written down. But nobody agreed to allow illegal immigrants to come and stay here.

As you acknowledge, the term embeds within it the premise that “the lack of enforcement of the law is also a kind of consent.” That is, illegal immigrants have America’s consent to be present in the country, because the government doesn’t try to hard to deport them. You can make that argument, but almost nobody would agree with your premise. So the term is smuggling an implicit premise that almost nobody would agree with if the premise were stated clearly (as you have done here).