My .02. Does rotating weight increase torque? yes and no. No, it does not create more power. That is all done in the engine. Yes, it stores rotational inertia so that it can be used later.
Probably a more appropriate analogy for a flywheel (or any rotating mass) is to think of it as a rechargeable battery. It doesn't create power, it only stores it for future release. You have to insert energy in order to retrieve it later.
A massive battery would take a lot of charging to get to a certain electrical potential. Same with a large flywheel. You will have to absorb a lot of energy to spin it up. Once spun up, it releases that energy back into the system. This helps smooth out the rotation between firing strokes, helps maintain energy when engaging a clutch, smoothing out the engine on overrun with the throttle off.
A light flywheel will "charge" faster, absorbing less energy. But it has less to release. That makes throttle response better, but in turn could make shifting balkier and clutch action more sensitive.