Regardless of
when you swing your arms, by doing so you're introducing additional energy into the system so is kind of beside the point. The relevant comparison (in the context of their paper) is
how fast you can convert chemical energy into kinetic through your gait, not
how much chemical energy you can load into the springs. Jumping stilts are a 4-step cycle:
1a. LOAD (Foot-ground contact, foot is descending)
2a. UNLOAD (Foot-ground contact, foot is ascending, optionally arms swing to boost upward force)
3a. WAIT (No foot-ground contact, foot is ascending)
4a. WAIT (No foot-ground contact, foot is descending)
Their proposal is 3-step:
1b. LOAD (No foot-ground contact, foot is descending)
2b. UNLOAD (Foot-ground contact, foot is ascending, optionally arms swing to boost upward force)
3b. WAIT(No foot-ground contact, foot is ascending)
Step 1b performs the same function as both Steps 1a and 4a, but is closer to Step 4a alone in terms of duration. By reducing the overall time one cycle of gait takes you're able to convert the same amount of energy from chemical to kinetic in a shorter amount of time (ie. achieve a more powerful gait cycle). This presumes that you can achieve equivalent levels of energy input with both methods, but this doesn't seem like a huge reach to me since both methods rely on springs so presumably you could fiddle with the spring constants until the overall energetics are the same.
It's kind of like how an 18-wheeler can produce tons of torque (which it needs to move something heavy), but has less horsepower than a Porsche. Torque is what determines how much weight you can haul (or in the case of jumping stilts, how high you can go), but horsepower is what determines how much time it takes to go from 0-60MPH. Swinging your arms is like adding more torque, but the paper is about maximizing horsepower.