So, throwing ideas to the wall here.
Within clubs, you’d have tonfas, you’d have escrima sticks, umbrellas, and canes, amongst others, and you’d have hammers.
Tonfas are kinda their own thing, there’s not that many weapons with sideways handles.
hammers, maces and the like have more in common with how they’re wielded with axes than with anything else. There’s not much difference in how you’d wield a hammer, a military pick, and a tomahawk.
The rest of the list honestly has more in common with swords out of all things, only adding a few possible moves like using them as aids in grabs and disarms that really better fits within martial arts than being contemplated within “skill”.
I suppose any system you pick is going to have it’s quirks, it’s weapons that blur the line. Even if you split by handedness you’d still have plenty of swords that can be wielded both one and two-handed.
I don’t have a good word for “axe-like” so you could dump all weapons that are basically a stick with a striking thing on the tip. Even if you could, you’d still get fuzzy things like falchions, kukris and machetes: swords and daggers made to be unbalanced like an axe, so your way of striking is kinda in the middle between both.
Then again, the current gun model has the same flaw since once you add a lot of weapon mods, the lines between pistol, smg and rifle get rather blurry. Even if you changed guns to work on handedness rather than type, you’d still suffer from this thanks to forward grips and stocks.
And where the heck would you put whips? Chains? Wind and fire wheels?
I guess there’s also the option of following the D&D model of simple/martial/exotic, perhaps with renaming martial with ancient or something. The advantage of this method is that you can make the training of exotic weapons skill slower in account of them being weird or difficult to wield. That way a chainsaw lajatang would take longer to master than a simple spear, even if both are polearms.