The goal of hangman isn't to minimize the number of guesses (if it was, your approach would make sense), but to minimize the number of wrong guesses. Or, to be more specific, have no more than a set number of incorrect guesses.

That actually means that the hardest hangman games are where you have to guess short words. Given that /usr/share/dict/words has 23 three letter words ending in at (only "aat", "iat" and "uat" are missing) only luck determines whether you "win" guessing a word like "mat", "cat" or "hat".