Not only that, but each arm can produce dozens of newton-meters of torque, with possibly sub-millimeter precision, and like you said, fast. Just building an arm with comparable specs is quite the challenge. You end up needing hydraulics because even the finest, most expensive transmissions with belts to move the motors closer to the body, still end up being too heavy to achieve human agility+strength.

I looked into building a robot arm which could cook breakfast. It's almost ludicrous just speccing out the torque requirements to pick up a skillet.

You end up having to pick harmonic drives and less speed but more stiffness, or hydraulics, which are lighter, but messy, less stiff (can be both good and bad) and less reproducible.