No, it scores you based on the number of moves used. No need for an upper bound, could've let me use 20 guesses if that's what it takes (non native speaker). But that wouldn't fit their copy&paste result formatting..
There is a UI for the game without the upper bound at connections.swellgarfo.com. Personally that annoys me as my friends are prone to sharing massive walls of incorrect guesses when they do badly on that site, but it sounds like it would be a good fit for you.
Too many people would take a scatter-shot approach to solving, even with a score that keeps track of guesses, and then those same people would get bored and disillusioned with the game because it would be boring.
The limited number of attempts is precisely what makes it a game.