If you've ever handled a fire axe, you'd realize it makes a lousy weapon. It has a long handle, with a heavy head. The idea is it can build up a lot of momentum to crash through things like doors and walls.
But the long handle makes it difficult to swing in a melee, and slow to swing, and once the swing starts it will be very hard to change its arc. Hence, your target can easily sidestep it. I suppose it would be good against plate armor, but not many villains wear plate armor these days. You also have to be careful with a fire axe to not chop your foot if you miss. I don't think I've ever seen a war axe/hatchet/tomahawk anywhere near that size.
For self-defense in close quarters, a baseball bat is ideal.
P.S. I am no martial arts expert.