Dungeon Treasure Coffers: Must all players in the party "right-click" on them?

[Rongway]

And it's likely I just missed seeing the "button in the corner of the screen"

The loot button will appear in the HUD element called "Notices". There's not a visible frame when you're playing, but that's what the element is called in your /hudlayout. It's the same area that contains pop-up notifications for party invites, friend requests, and returning after getting knocked out.