[Update] I bought a drive kit and installed it.
You’re looking at the starter for a Kohler SV735-0016 twin. For those not familiar with this starter, the small “washer” below the gear is threaded on the inside. When the starter is energized, the shaft spins counter-clockwise and the washer pushes the gear up to engage with the gears on the flywheel. There is no direct connection between the starter shaft and the starter gear. Instead, there’s supposed to be a rubber piece that is smooth on the bottom and slotted on the top to mesh with the gear. It transfers the rotational force of the washer to the starter gear. The remnants of that part can be seen in the picture.
Age and heat did it in. We knew something was horribly wrong by the screeching noise it was making when it was last started. The rubber piece had split, managed to get wedged between the washer and the starter body, and was holding the gear against the underside of the flywheel. That’s why the top of the gear is somewhat polished. There’s also supposed to be a cap on the top of the shaft which contains a spring to push the gear back down after starting. Clearly, that’s missing too.
Now I have a conundrum. I can buy an OEM starter for around $180. I can buy an aftermarket starter for as low as $70. Or I can go really cheap and buy a drive rebuild kit for about $25.
I hate to see everything replaced when all I really need is the rebuild kit. However, replacing the whole starter is simpler and guarantees that it will work.