While you CAN use Warfare skills with a Staff, it runs afoul of not being able to break Physical Armor for the various effects like knockdown and the like. So it's not AS good as it sounds.
Some skills in Huntsman, Scoundrel and Warfare don't care about your weapon equipped. Adrenaline can be used with any weapon (as an example).
Most Warfare skills require either a melee weapon (which includes staff and dagger) or a shield however. Just like most Scoundrel skills require a dagger and most Huntsman skills require a Bow / Crossbow.