They used to, but no longer ever since 1.6 Weapon/Spell Power normalization.

You want Spell Crit for Staves, and Weapon Crit for everything else. In general, if an ability costs Magicka to activate, it will scale off Spell Power/Crit. If the ability costs Stamina, it will scale off Weapon Damage/Crit.

In general, if it looks like a spell, it goes off Spell Crit/Spell Damage. Otherwise, it's Weapon Crit/Weapon Damage. Stave light/heavy attacks fall under the catagory of "Looks like a spell", so ZoS made then scale of Spell Damage/Spell Crit.

Staves used to scale off Spell Crit/Weapon Damage back in pre-1.6, but that confused a lot of people, so it got normalised in 1.6.