[Perdition]

Currently, base duelist is unable to use axes as a weapon inherently and also possesses no skills specific to axes whatsoever. However, when you upgrade to Ghost, axes have the most skills specific to them, even if only by 1. Axes have Dark Imbue, Cut Down, and Reaper scythe compared to sword's Sanguine Star and Ghastly Strike. 

Despite Ghost's axe skills (aside from Dark Imbue) being very strong, and arguably the best skills for axes in the game, it feels awkward to use because your base class has no interaction with your weapon. You're left with many points that are useless if using an axe as a ghost. 

Conversely, Ghost cannot use spears inherently and has no skills that are exclusive to spears. For at least Duelist, it would be nice if a line of skills were added specific to axe so it doesn't feel as bad to play Ghost and have almost no synergy from your base class. Magic Gunner suffers this somewhat when upgrading from archery, although at least some of the bow skills there are gun compatible too.

A case could be made for Ghost having spear skills as well I believe, although that's not as important as base duelist getting axe synergy. While Firebird and Kensei exist as specialist classes for spears and swords (katanas specifically I guess) respectively, Ghost was never tied to one specific weapon out of axe and sword. Having it be the upgrade that uses all of duelist's weapons by incorporating spear skills to it would be nice QoL in my opinion, especially since some spears play very nicely with ghost otherwise such as Fellgrant Drinker.

[Snake]

I'd just make Sword skills be able to be used with Axes for idk, bonus critical rather than hit.

[Autumn]

Sidecut and Sillcut should be generalist skills for all of Sword/Axe/Spear most likely, while the other more specialized skills can stay for Sword/Spear

There could be a couple of Axe Skills in the class after doing that most likely, but I think thats all it really needs to push it forward even a little bit.

[WaifuApple]

I suppose I don't have too much against this if it gets axes more love in this direction. It feels weird to go from no axe support to sudden axe support in a promoted class.

[Shujin]

Especially since axe is supposedly the Crit weapon.
I wouldn't mind side/sillcut being generalized too much, but it would probably blurr the thin line we have between weapons... But I am not entirely opposed to it, simply because of how thin the line actually is.

[Lonestar]

Making Sidecut/Sillcut also be casted by Axes would lower the reliability that Axes have on BK, due to Hanging's mobility. Not to mention it's a particularly generic skill that I don't see why Axes could not be able to use it.

Gives some love to the weapon type (while a promo focused on that weapon alone doesn't come).