Natto learned “Fireball” it was super effective
Also yeah that’s a classic misconception, doesn’t have to be a staff, just a medium you’re attuned to. Magic swordsmen are a thing after all
Well, it depends on the setting.
How well can the item you are using conduct the magic power. Or how it may help shape the spell in making it easier to convert the raw mana into the desired effect.
Wands and staffs are usually the ones with the best properties for this.
Swords (of a suitable material) or most often only the conducts, but without the spell shaping help. Thus why magic swordsmen usually are weaker than pure mages who use staffs purely in regards to the strength of their magic effects (but making up for it with their swordsmanship added in).