[Tuor Turambar,Cursed by the Valar]

Actually, I read the concept of rebirth in either the Sil, or The Book of Lost Tales. I don't like the idea, because it is too similar to the New Age/Eastern religion and mystical ideas of reincarnation. But the book stated that Elves are reborn in their children somehow.

Yes, I know this, but:
I do not see a single example of elvish rebirth in any of the books I have read. It says that it exists, but never states a specific example. Because of this, I have3 come to believe that it was an idea that he toyed with but then discarded. I even read something saying he never used it in HoME somewhere.