> You are not a brain on a stick
We probably are not, but I'm not convinced that the opposite is true.
Stephen Hawkins did not notoriously exercise much, given his condition, and yet he died at 76, despite the odds.
Einstein routinely walked a few miles a day and that's all the exercise he did.
Many other popular great thinkers of the past spent a lot more time on books than improving their bodies.
In Italy we learn as children about Giacomo Leopardi, a very popular poet, writer and philosopher of the 19th century, according to Wikipedia "considered the greatest Italian poet of the nineteenth century and one of the most important figures in the literature of the world", that His continual studies undermined an already fragile physical constitution, and his illness, probably Pott's disease or ankylosing spondylitis, denied him youth's simplest pleasures
Without evidence I reckon that all we can conclude is that only sometimes exercise is the best cognitive exercise, but we don't know why nor if it is actually true.