While I agree with what you say, I'm not exactly certain how you'd program exercises that are pretty demanding into a cooldown. For warm up I suppose it could be done, but they dont exactly have a large movement component that would prepare the body for motion. I believe you might be missing the fact that a pullup and muscle up–both of which can (and often should) be programeed into the body of a strength workout–are gymnastics exercises. Exercises like hand stand pushups (albeit after some progression), and hollows can fit well when programmed into general strength circuits.
when i say cooldown this is what i mean: i would do speed and quickness in morning then come back and do my strength P.M. my cooldown for strength session would be core and quasi iso lunge on my lower body days. core work would be 10mins of work of the following drills; physioball twist crunch, cartwheels, speed russ twist and somersaults. you should already know i feel about such topics like this.