Again, are you absolutely certain it is full on oil. The oil sensors on this older B&S engines required to oil to be absolutely full to reset them, maybe a 1/8" over full.
B&S came out with these oil sensors in the 80's when I had my shop. I can not tell you the number of engines that came in that were a touch low that didn't run. Toss in a few ounces of oil and they would run.
I had a large construction company with Lincoln welders that were powered by 16 HP B&S engines. They would come in with the connecting rod broken, the oil was extremely low and the low oil safety disconnected. They kept telling me the low oil safety was bad, but I was certain they never filled the oil all the way. They would tell me they put some oil in it and I would ask if they filled it? Nope! Usually they put in enough to get it half way between the add and full, then snip the wire because it wouldn't run! I would rebuild the engine, reconnect the low oil shut down and it worked just fine. I preached to these guy to not just sort of fill it up, but over fill it slightly to reset the safety.
For what it is worth, I never saw a failed low oil sensor. They don't fail.