The experiments on teaching sign language to chimpanzees actually suggests the opposite, because it shows that, even if you take the physical inability issue off the table, the chimpanzees do not make much use of their language skills - they do not enter into extended conversations with their handlers or one another. The evidence that they 'truly' (perhaps we should ask 'how deeply') understand language is controversial, especially given the recent evidence as to the extent that computer programs can give the superficial impression of having more understanding than a more thorough study of their abilities reveals.
It is sometimes argued that we do not know enough about their motivations to make that call, but it is hard to imagine there being no situations in which they would find language useful, and even harder to imagine not being able to devise experimental situations in which a greater use of language would be the only way to achieve goals they manifestly desire. Ultimately, the argument about different motives is one about different mental capabilities, anyway.
Personally, I think it probable that species other than Homo Sapiens made good use of language, but we seem to be the only one around at the moment.