And this one has special rules for being broken, none of which mention having to be attuned to trigger it.

"Retributive Strike" is a magic property (just like 'Power Strike' and its Spell List). You can't activate or benefit from magic properties on non-attuned Magic Items/Artifact.

So if you break the Staff of Magi without using that Action, well it becomes just a broken Staff.

Also thanks for teaching me that one can indeed can break magic items, that non-destructible nature only applies to Artifacts.