Why can't you meet yourself?

You said in the question "what you've done in your 'past' still happens (dungeons being done and stuff)," but this is untrue. When you play the Song of Time, everything reverts back to the first day: people get un-rescued, rupees get un-collected, dungeons get reset. The only things that stay are the items, like the bow and things (which we can assume are magic) and the markings on Link's body (e.g. the stamps he gets to save rupees in the bank).

Another answer to this question would be that this would take a ton of storage space to save all the movements of every single Link.