I think the Tree of Life and timeshift stones are the exceptions to the overall timeloop that is SS rather than the Tree of Life and timeshift stones proving the game isn't a timeloop.
The Master Sword follows the same mechanics as the Tree of Life and the Timeshift Stones, and it's a plot thing. It's not that easy.
I believe the Master Sword to have been there all along, and old Impa was aware of it. We already know the pedestal the sword is put into in the past was initially invisible until Link must put it the sword rest in said pedestal, so I believe the whole thing is hidden so no questions are asked when Impa tells Link to get the Sacred Flames to make the Master Sword in the first place.