Below is a listing at TM Tractor for a new starter switch, if you need one. The contacts burn after years of use, then it doesn't work anymore.
You could sand the contacts with some sandpaper, and clean them up, and maybe the switch will work, sometimes they are too burned, and need replacing.
Be sure the switch has the 2 insulation pieces shown in your parts pic above in place, at the sides of the contacts, they are needed to keep it from shorting.
Disconnect the battery ground before removing the switch.
http://www.tmtractor.com/new/el/2543fp.htm
You could remove all 4 battery cable connections, one at a time and sand them, and where the attach to, and clean them, they commonly get corroded, and don't work good. Wipe off all sanding with a clean rag.
It takes some effort to turn the engine, with it in 3rd gear, and turning a rear tire.
You could see if the engine will turn any, turning the fan, with the transmission in neutral.
Below is a pic of the IH carburetor showing main jet, it is the brass colored bolt head on the lower side of the carb.
The 2nd pic shows cleaning the jet, wash it too is helpful.
You could remove the little drain plug on the bottom of the carb instead, it will let gas run out too. Tighten parts gently on the carb, they are soft metal, and threads strip easier than threads in cast iron.
