The problem is that there isn't a single key in this case, there are many. If there was just a single key to begin with, fuzzing would not be very fruitful for

The problem is that there isn't a single key in this case, there are many. If there was just a single key to begin with, fuzzing would not be very fruitful for any moderately+ complex program since finding that single path would already be very hard (and therefore has a high cost/benefit). What fuzzification actually enables is to hide more keys (crashes) from a constant amount of (current) effort. Those crashes are still there and now there are more of them, waiting for someone to find them with an alternative method or simply by buying more fuzzing equipment to scale horizontally.