Lose Track does apply, but it is important to remember that Lose Track never stops you from playing a card, it only ever stops a card from being moved.
The steps that happen (skipping over the steps you know such as getting +cards and +actions)
1. Play Throne Room
2. Play Mining Village; it moves from your hand to in-play.
3. Trash Mining Village; it moves from in-play to the trash.
4. Play Mining Village again. It cannot be moved because Throne Room has Lost Track of it. It can still be played, but it stays in the trash. So you still follow the instructions on the card because you played it; it just won't move from the trash to in-play.
5. Try to trash Mining Village; you can't. There are actually 2 separate reasons that you can't. First off, because you can't trash something that's already in the trash. Second, because Lose Track applies. Mining Village expects itself to be in-play, but it isn't. Because it isn't where Mining Village expects, it can't be moved.
Because you never actually trashed it the second time, even if you tried to, you don't get the +
the second time. The "For" wording requires you to do the first action to get the benefit.