Although you are right to be skeptical of this claim because near-native competency is always possible, L2 competency (learning a language as a foreign language) will always be less than L1 competency (native). For instance, there are grammatical constructions with very low input frequency about which native speakers have strong intuitions, which L2 speakers will not have. The difference in L1 and L2 intuitions may be remarkably subtle, often semantic/pragmatic, but you don't have to work in the linguistic community for long to come across them.

I vaguely recall watching a program on tv in which there was an African tribal language composed of clicking sounds and other sounds that were claimed to not be able to he repeated unless learned from birth. I'm not entirely sure of the specifics but I remember it being cited as an "unable" to learn language

More likely, no one cared enough to make the effort to become fluent in the language.

I would say that if it can be categorized as a language, it can be learned, because categorization would require the observation of patterns of cause and effect.

(I know they're illusions, but they're useful illusions.)

Then it's just a matter of gathering data, developing hypotheses, and testing your interpretation of the data by asking someone if their hovercraft is full of eels.