First of all, the theory described in the article does not even believe that all humans are conscious, so it does not need to account for consciousness in other animals.
Secondly, my point is that other animals may still have "language", but just without the small components necessary for communication. If this were true, then language could still be the explanation for consciousness.
It's not. That's why I said that any good theory is going to have to account for all the sentience in the world that doesn't talk to each other.