Illegal immigrants aren’t “undocumented.” They are present in America without the consent of the American people, as expressed in immigration law.
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).
In 90% of cases somebody did. Somebody at the border talked to them. Somebody checked their passport or their documentation (at the border). So many "undocumented immigrants" are people who overstayed their visas, like Elon did.
They literally were documented but then they didn't keep up with the paperwork and thus became undocumented.
"Doccumented" status *is* literally just a case of keeping up with the paperwork.
Some people do sneak in but the *vast majority* come through legal ports of entry
"The allegations are undocumented."
In the context of immigration, one could say "she claims to face persecution in her country of origin, but this is undocumented."
That doesn't really apply to "undocumented immigrants," because it's usually not in doubt someone is an immigrant.
If their place of birth is unknown, and might be in the United States, you could use the phrase "undocumented immigrant."
I agree it's a propaganda phrase, and does not conform to clear English usage.
But "undocumented" does not mean something is sanctioned, it means it must be proven and has not been proven.
