Welcome to the group Makis. Not sure what the forward winch (probably what is properly called a windlass) was used for , likely handling mooring lines or help in raising sails and yards on the mast. The capstan would have been used to raise the anchor due to the weight of the anchor as it allows more men to operate it. If they had to walk over the gratings they would have, possibly the grating would have been covered with canvas giving better footing. Also the capstan was used with long capstan bars to give greater leverage so they may have missed the gratings when working the capstan. Unfortunately most of my references are for naval ships and the Bounty was basically a merchant ship taken into the navy.
Jim