> Brains do much more than output language.

Hmm, what do you think about? Unconscious control of the body's processes? REM-phase dreaming? Reaction to hallucinogenic substances? Automatic actions of trained fighters (soldiers or martial arts practitioners)?

Are they critical to distinguish actions we attribute to humans from "non-human" ones?

I don't see any human activity not directly, or at least indirectly but closely linked to the use of language.