Except in the case of teleportation, one of them is stopped after the other has started. For ease of making my point clearer, lets say it takes a few seconds after teleportation to destroy the original.

For those few seconds, there would be two divergent consciousnesses. The original consciousness would not experience the consciousness stream of the copy. It would be left experiencing the inside of the teleporter, and then it would be extinguished. The copy would have access to the memories of the original consciousness, and would infact experience itself as a continuation of that consciousness stream, as would anyone and everyone that interacted with the copy.

But the original consciousness, the one that was copied, briefly existed simultaneously with its copy, yet distinct from it, before being extinguished.

Even if you believe that every moment of life is some version of that, where our experience of continued consciousness is not real, where we are "reconstituted" continuously as new versions, with only shared access to memory letting us perceive it as continuous, the teleporter still creates a second stream simultaneously with the first, before ending the first. You have a sense of self that is consistent and continuous. Even if you are recreated constantly, that is not how its experienced. You still fear death, injury, sickness etc, because you perceive those things as impacting you and your future experiences. You place value on your perceived continuation. And the teleporter breaks that, because there is no longer a perceived continuation for the original, only for the copy. And unless the act of copying spreads perceived consciousness across both streams simultaneously, one stream is going to experience its end.