Even if behavior was "embodied", wouldn't the brain notice? YES, of course: then the BRAIN would become the better "vehicle" for remembering, thinking, and "time travel" (i.e. prospective memory) -- possible (and possibly trivial) sensori-motor components notwithstanding. [ I am really quite tired of the "embodied" conceptualizations (which have yet to be shown as non-fictions *). See my writings. No one has argued against the views/approaches (content) in these writings NOR accepted/liked/or adopted them (now 1+ years (or 5+ years, depending how you look at it) and counting). ]
* Footnote: All this nonsense is ALL because NO PSYCHOLOGY OUTLOOK (other than my own) "believes in" anything psychological, innately guided, and emerging with ontogeny (which is not tenable). (The idea that learning is literally nearly always "the same" (outside of clearly always being associative in nature) is preposterous (think of a two -year-old and an adolescent -- and imagine any systematic and universal instruction you credibly might posit). P.S. Relatedly : "Culture" does NOT directly impinge on the individual -- the actual Subject and ultimate, but absolutely necessary, unit of analysis &/or explanation (for Biology or for Science). All executive or "meta" processes can NOT be properly shown to be anything but homunculi.)
https://www.researchgate.net/project/Human-Ethology-and-Development-Ethogram-Theory-A-Full-Fledged-Paradigm-for-PSYCHOLOGY