Would Brink of Disaster destroy an already tapped permanent?
It's all in the rulings:


  3/1/2010  Brink of Disaster may target and may enchant a permanent
  that's already tapped. It won't do anything until the enchanted
  permanent changes from being untapped to being tapped.
  
  3/1/2010  When the enchanted permanent becomes tapped, Brink of
  Disaster's ability triggers. That permanent will be destroyed when the
  ability resolves, even if Brink of Disaster has left the battlefield
  or is somehow enchanting a different permanent by then.
  
  3/1/2010  If the enchanted permanent is tapped as a cost to activate a
  mana ability, the mana ability resolves immediately, then Brink of
  Disaster's ability goes on the stack.
  
  3/1/2010  If the enchanted permanent is tapped as a cost to activate
  an ability that's not a mana ability, Brink of Disaster's ability will
  go on the stack on top of that activated ability. Brink of Disaster's
  ability resolves first (destroying that permanent), then the
  permanent's activated ability resolves.


These rulings are all clarifications that follow from the card's own rules text:


"When X, do Y" is a triggered ability, meaning that the action will go on the stack after the triggering event occurs. If you tapped something to use its non-mana activated ability (like Steel Overseer's), then the effect is still on the stack when Brink's trigger goes on the stack -- on top of it.
The trigger text is "becomes tapped". It doesn't care about the current state of the permanent, only when a specific action occurs.