Reading though a book last night, the author presented the idea that in Gait, the deltoid acts more as a stabilizer then as a mover. The Arm swing is more to provide a shock absorbing system almost to allow the neck to be a stable platform.
The deltoid is an interesting muscle with its pennate fiber angle. It is active in the forward and backward swing of the arm, but is active because of, not to create it. It absorbs the rotation of the body to help keep the head level.
These ideas are based off research by Pontzer, Holloway, Raichlen and Lieberman. Control and function of arm swing in human walking.
I often find that patients have a hard time really engaging the deltoid and want to shrug or flex the biceps. The Delt is a direct continuation with the trapezius, so it makes sense. Also, the trapezius contracts just prior to heel strike in the gait cycle. Another mechanism to help keep the head level and not allow the head to pitch forward.
Cool Stuff. Reading through Born To Walk, by James Earls.