I'm really skeptical of claims like « the "thinking unit" is really the whole body », they tend to discard quantitative considerations for purely qualitative ones.

Yes, the brain is influenced, and influences, the whole body. But that doesn't mean the whole body has the same importance in the thinking. The brain is also influenced by lots of external factors (such as ambient light or sounds, ...) if as soon as there is a "connection" between two parts you say "it's the whole system that does the processing", you'll just end up considering the solar system as a whole, or even the entire event horizon sphere.

There is countless evidence that, while your body and your environment have significant influence on your thinking, it's just influence, not fundamentally being part of the cognition. For example, people who have graft or amputations rarely change personality, memory or cognitive abilities in any way comparable to what brain damage can do.