Originally posted by: Gaffer Tape
I've actually been thinking about this, because you made me think about it in a way I hadn't considered before. But ultimately I realized why it wouldn't be able to work. Based on the book (and even more apparent in the movie) is that Rowling's view of a time travel is a fixed timeline, unlike, say, Back to the Future, where you can actually change events. For example, Harry saving himself and Hermione with the Patronus in all instances of that timeline, not just when he went back and did it. If Voldemort could have been stopped that way, then it would have already happened, and it obviously didn't.
More importantly, and probably logistically why no Auror would try it is because it would create a horrible paradox. If they did destroy Voldemort in the past, there would be no reason in the future for them to go back to the past and attempt it, which means that no one would, so Voldemort wouldn't be defeated, etc...
I've also been thinking about this, and it seems to me that the view of time travel you described leads to another huge paradox. If it's true that you can't go back and change something unless your future self already did, that implies that there is only one timeline in existence, and that timeline is predetermined to a huge extent if not entirely. But free will plays such an important role in the series -- just think of how many times Dumbledore tells Harry how important his choices are, and that the prophecy could have swung either way (Harry or Neville) until Voldemort chose to "mark Harry as his equal", etc. These kinds of choices can't logically be freely made in a fixed timeline.