No Dark Elf, Elrond did not leave Middle Earth while Frodo was on the quest. And the question you asked is a good one, and certainly food for thought. The only thing I can think of right off the top of my head, is what someone else said in another thread about Elrond being a higher elf, and so sauron might have spotted him and recognized him and his intention because of it. And since the fellowship was attempting to travel in secret, I suppose it wasn't the best idea to draw unneeded attention.
Something will hit me later and I will get back to you.
Later,
Leona
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You don't really see me, just the cloak of shadows and lies I've drawn around me.
-Arina
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