The other thing I suspect is that "Just give me True/False" cuts off a large amount of the search space a modern-day LLM uses to help it answer questions (you can see it in reasoning traces but the act of writing the explanation helps guide it toward a better answer and gives it better likelihood it backtracks on a bad decision).

If you let it spew out an explanation along with the answer, I'm curious if the accuracy will improve (I suspect it will).