I'm not that familiar with Kearsarge, but photos like yours indicate that the anchor-capstan set up was typical of sailing ships of the 19th Century.
The anchor chain ran up into the ship through a hawse hole that led to a hawse pipe. The chain in this case came up into daylight, under the little forecastle deck, onto the main deck and ran back to a fitting called a spurling. That is the part you drew arrows around. Where those were located is beyond my knowledge, but close to centerline makes sense as they had to pass through the deck below to get to the lockers, and those would be side-by-side.
Passing into the spurling, it would run down a pipe called either a spurling pipe or a chain pipe where it fell into the chain locker.
The chain(s) never got close to the capstan. The capstan pulled on a stout loop of rope called a messenger. That had a number of turns around the capstan and usually ran forward and looped loosely around a bollard or post that kept it positioned fore and aft. It also lays at the feet of those gentlemen in the photo.
The strain of the capstan messenger is transferred to the anchor chain by nips or nippers, usually short pieces of line that are used to lash the messenger to the chain as it comes aboard and removed at a point where the chain and messenger diverge in direction.
Once fully weighed, the anchor chain is secured with a stopper, a short length of stout rope or iron bar with a hook on each end. One end is attached to the anchor chain and the other to an eye attached to a stout timber on the ship.
As for Kearsarge, as I said, my knowledge is minimal. I hope that helps.
Bill