An overhand pitch actually requires much more agility than an underhand pitch does, I assume. It just makes sense. Overhand pitches call for heavily linked and connected activity involving a lot of force from the rest of your body (back, pelvis, and even the other arm).
Try pretending to pitch overhand... See how much force you have to exert?
Now, try underhand... It barely requires excess force to achieve, doesn't it?
Underhand is simply more natural because overhand demands your arm to somewhat bend in on itself and almost exceed the shoulder joint's limit, which is why you tend to hear more about injuries resulting from overhand pitches, not underhand pitches. We try to exhaust the least amount of energy possible.