I have a B&S model 204412 on a troy bilt generator. The engine stop/run wire seems to be grounded all the time, even with the actual switch disconnected from the circuit. Obviously the engine shouldn't run in this configuration, or should it?

Hi,
It is likely that the Oil Minder is grounding it out. That is the float device on the right side of the engine (looking from facing the starter) that prevents engine failure in a low oil situation. Checking the oil level is probably called for. Usually it ai a yellow wire that goes to the Oil Minder. It can be disconnected to test, though it should not be left disconnected unless you just like to buy engines or short blocks.
Peace,
Clarke