Perhaps an overstatement, here?
Other suggestions:
"does not always"
"need not"
"can [exist] in other places besides"
And then we have the whole problem of "instinct" e.g. see some accounts in When The Air Hits Your Brain by neurosurgeon Frank Vertosick, where he recounts removing most of the brain of cats, and the reactions of the cats' owners afterwards that "they couldn't tell the difference". In other words, much behavior may be ingrained in muscle memory or elsewhere, without the brain being required to perform the action--but the brain may still play a role in when, how often, how extreme the action is performed, and in regard to what stimuli.
In other words, how sure are you that the situation must be an "either / or " ??
Cheers! [Full Disclosure : Just stirring the pot]
I would add however that where the autonomous organism (like the cat) is comprised of molecular machinery (cardiovascual, neural, digestive, etc.) integrated to the purpose of satisfying the organism's will to live --- that the brain is the most frequent (but perhaps not exclusive) site of the origination (inception or beginning) of the cascade of information (successful communication) throughout the organism to accomplish its will.
This is based on the Shannon "mathematical theory of communications" - which is the bedrock to "information theory and molecular biology". Shannon's theory is the basis of the field, "information theory".
Information is the "reduction of uncertainty in the receiver or molecular machine in going from a before state to an after state". It is the action, not the message. The value or meaning of the message is irrelevant to the theory.