I think Mario could also be seen as someone to lead with jumps, which are actually kinda haxy in some games. In Mario RPG, he gains the special to jump before anything fireball based if we appeal to RPGs, and it's more-or-less his go-to in the proceeding RPGs. In games such as Mario RPG, Super Mario Bros 3 and possibly Super Mario 64, his jumps are seem as more powerful than his punches (for example, in Mario RPG, punches are standard moves and jumps are specials, in Mario Bros 3, his punches require a Super Mushroom to break a brick block, whilst even as small Mario his jumps can break them, etc), and considering how versatile his jumps are, with numerous techniques, it makes sense for him to want to use them. Plus, in basically every platformer in
the mainline series it's his standard move and general go-to (of course, with some exceptions, but it's still there for him). Since he can change the effects of his jumps in some RPGs, such as making a spiked ball occasionally fall or making it so jumping on a spiked foes has no ill effect, not to mention his unenhanced jump seems to drain foes of their paint/colour, which appears to be their life force (such as how a colourless Toad became a lifeless envelope, or the game over screen being Mario drained of his colour), you truly can't blame him for being Jumpman. (not to mention his possessed body did this).
So, whilst it's an appealing argument to say he leads with fireballs, I personally think jumps are also a very good candidate.