What we need is a large corpus of vocal recordings along with contextual video recording. Then train a model to associate vocalizations with states of the video by predicting subsequent video frames. By throwing enough compute at the problem we could probably come to understand any animal language with such a technique.
Looking at correlations between vocalizations and behavior is one thing primate researchers have been doing for decades. It seems implausible for there to be a large skill set that has so far escaped their notice.
Whenever one group of humans has met another, it has rapidly become apparent to both parties that the other has a language, no matter how long the ancestors of the two groups have been isolated from one another. Given how long humans and chimpanzees have been coexisting, if they had comparably effective language skills, it is implausible that either party has not recognized it in the other.
I think its been conclusively demonstrated that deep learning can discover correlations that go unnoticed by humans. An example that immediately comes to mind is AI detecting race from X-rays. There are many others. There's also many other factors that influence our ability to ascribe language skills to other groups of humans, from complexity of vocalizations to similar ways to interact with the environment and with other group members. We recognize ourselves in fellow humans, which makes it easy to ascribe similar capacities. The question is how easily can we recognize language in a species where we do not immediately recognize ourselves? The more foreign the behavior and the patterns of communication, the harder it gets. AI has the ability to elide over those difficulties.
I do not doubt that deep learning can discover correlations that go unnoticed by humans, and if our failure to detect the level of language skills we possess in chimpanzees (and the failure of chimpanzees to detect those skills in humans!) is due to the sort of difficulties you raise, then I agree that machine learning could find the evidence that we have not. What I do doubt, however, is that these difficulties actually exist to the level required for us to miss what is going on.
I believe that the theory of evolution is correct, and therefore that the evolution of language confers a differential fitness, yet chimpanzees are clearly not behaving in a way that exploits anything like the full capabilities of a human-like language, either in interactions between the members of their own species, or with their adversaries such as leopards or humans. At any time in the past, a small cabal of language-wielding chimpanzees, who grasped something more of the capabilities of their language skills than did their neighbors, could have dominated the remainder, increasing both their fitness and that of their species. In practice, we do not even observe them chatting.
It was David Brin, I believe, who made the same point with respect to dolphins: if they have these language skills, how come the knowledge of the dangers presented by purse seine nets, and the means to avoid them, has not spread within those dolphin communities where these nets are a significant cause of mortality?
There are many cases in evolution where a feature evolved for one purpose but then served another (which is how, it is supposed, birds got their flight feathers) - but even if that was the case for chimpanzee language, what was that primary purpose and why have the abilities not been adopted for their secondary purpose - especially as, once you have that level of language skills, change occurs at the rate of meme spread, rather than that of genes.
In other words, the combination of human-like language skills and the chimpanzee lifestyle seems very far from equilibrium. Language, by its very nature, does not deliver its evolutionary benefits cryptically.
The view I set out here is neither predicated on nor entails the position that human language is unique. In fact, I happen to believe that there have been several species on Earth with human-like language, it's just that the others are now extinct. I also think it far more likely than not that there is language-using life elsewhere in the universe.
I expect there's a hierarchy in language acquisition, where describing interactions with the world come before more abstract talk such as gossip. But once this first-order descriptive talk is understood, this can be leveraged to capture the more abstract talk.
