The main benefit of physical exercise to me, is to smack the autonomic nervous system / ANS into shape. Conscious computation must handle work that isn't subconsciously managed by the ANS ... over time, as conscious computation manages and adjusts tasks originating from the ANS, conscious resources are more and more limited. Performance of conscious computation is strongly dependent on a successful regime of delegating work back to the ANS.
Basically, modern life is not what the body was designed for. The ANS basically gets lazy and disfunctional, and needs to be tricked, into feeling like it's still out on the savannah punching tigers. To explain this more basically ... ploughing fields, or moving hammers at speed, are equivalently stressful.
This is a good consideration for doctrine.