I'm still slightly confused about this, even though I thought I understood the Lose Track Rule.
Maybe we can clear this up with some hypotheticals. If Throne Room told you to "play a card, trash it, then play it again", would that Throne Room move it from the trash to the play area the second time, or would it have lost track?
(If it would have lost track, then why doesn't Prince lose track of its set aside card?)
On the other hand, if Necromancer didn't have the "leaving it there" clause, would it move the card from the trash to the play area?
My understanding of the Lose Track rule is that a card can't be moved by an effect if it first leaves the place where the effect expects it to be. I think the part that's tripping me up is determining where a card is "expected" to be; would the Throne Room variant above expect the card to be in your hand the first time and move it to play, and then expect it to be in the trash the second time and move it to play? Would the hypothetical Necromancer expect the card to be in the trash and move it to play? (Also, in both of those cases I assume you would put the card into your discard pile at the end of the turn as usual, effectively gaining it in the case of Necromancer.)