Don't most mammals have about the same number of joints? This implies a very small part of our mind is responsible for that.

Even pumpkin toadlets, who famously can't coordinate their landings, the problem isn't the size of their brain, it's their vestibular system.

I'm of the opinion that most animals are probably a lot higher up the sentience scale than is the current social consensus, but I don't think exercise is itself a major cognitive effort. Arbitrary games sure (not just the rules, anything social involves modelling multiple entities like yourself), but not the physical exercise itself.