To add extra light on this... The text on Exodia's card is not an actual effect. It's simply there to state what the win condition is. You may not respond to a win condition because it has no spell speed. It is simply a condition, and thus your opponent automatically wins. Activating Trap Dustshoot on the Draw Phase that your opponent draws the last piece will not stop it because Trap Dustshoot doesn't resolve successfully before your opponent wins.
Extra clarification on win conditions: Think of it like drawing a card when you had three cards in your hand. Having three cards is a CONDITION, and once you draw card four, having four cards is a CONDITION. So drawing into piece five is a mandatory game mechanic defined by the Draw Phase, so you cannot stop it. For Destiny Board, although you can easily end it with Dust Tornado/MST/Heavy Storm, each letter is PLACED by the effect of Destiny Board. So if you plan to stop FINAL/DEATH with Solemn Judgement/Dark Bribe, you negate Destiny Board because you cannot chain it to a card PLACED on the field.