Thats a beautiful survivor! I have a ST 16 and my grandpa had two SS 16 twin with all the availible attachments but one was stolen. good luck with yours! I know them sears dont like to be low on oil.
Separating the Kill switch from the Ignition switch is a very very very good idea on the old Tecumseh engines. All it takes is a bit of moisture or wear or even a bit of dirt inside of the switch to Kill the Solid State ignition coil on those old Tecumseh engines. Any current fed back through the Kill Wire will instantly kill those coils and they are very expensive to replace IF you can even find a good used one. Leave it hooked to the headlight switch or wire it to a toggle switch.
Most likely the previous owner had 1 go bad. Every running Tecumseh HH100-HH160 powered tractor that I own got the same treatment.