This is one of the many reasons there should be floating perks for each class not attached to any tree that just have a level requirement not a skill reqirement. Mage could have had one for staffs and one for cloths/robes/no armour. Thief could have had acrobatics/athletics perk and a martial h2h perk. Warrior gets carry weight perk and resitance perk. Somthing like that.

Does inflicting damage with staves level up any skill? One-handed or destruction (or whatever the staff's corresponding school is), anything like that?

They give you access to spell effects a) if you're out of magicka or b ) if you aren't at a high enough level for a certain spell.

The only one I use regularly is Sanguine Rose. I'll occasionally use Staff of Magnus to sap an enemy mage's magicka.

Staves are amazing.....

I have a Frost Atronach staff, and my follower a storm atronach staff, and in hard fights it helps me keep a summon up even if they keep getting 1-2 shotted without draining my mana pool.

The Staff of Magnus is a nice lil mana refill staff for mage battles.

the Jabberwock is just entertaining.

Destro Staves are good to let you regen mana while keeping up the offensive.

It's one of the few ways to LEGITIMATELY level Enchanting. ( I enchant weapons I loot, and jewlery I craft/loot. I don't craft daggers for enchanting. Refilling enchanted weapons is part of my leveling system)

This^

The one I carried with my first character was a staff of chain lightning I picked up in a quest from some bad guy (I forget who). It was invaluable in beating one of the daedric quests where you're confronted with the summoned daedra at once. I got killed like 5 times before giving it a shot and it took care of them nicely. You can use a spell for the same thing I guess but in that situation your magicka would have to be through the roof to get the job done unless you had a ton of potions.

I thought the Wabbajack was cool at first but the fact that you don't know what it's going to do makes it next to useless in battle. Turning an icewolf into a chicken was funny but against three angry Falmer, not so much.

Right now I carry a summon daedra staff but haven't used it.

One thing I wish Besthesda allowed in skyrim was the user creating his own staff. If I remember right Oblivion kinda allowed you to make your own personalized staff but in this game it's impossible to. Which is disappointing