It seems to me that a trivial thought experiment makes it pretty clear.
Let's say we've got our two ends to the teleport device, A & B. Somebody walks into A, is destroyed and read into a computer, and is reconstructed at B. This is essentially what the original question is about, right?
Now let's say that some genius discovers a way to copy all of the person's atoms/whatever without destroying the person. The person walks into A, is copied, and an identical copy walks out of B.
Does person A have conscious experience of person B? Do they suddenly start seeing double, experiencing two sets of senses and two thought-streams, etc? Of course not, that would require something metaphysical, a soul or some such.
Sure, person B has the experience of continuity with person A, that is, they believe that they stepped into part A of the device and immediately stepped out of part B. In their subjective experience, they are the original person.
But person A, who originally stepped into A and then back out, is not experiencing what person B is experiencing. Now let's say that the moment they step out of A, someone stabs them through the heart. Or vaporizes them with a laser gun or something. Whatever, point is they're dead, and their consciousness ends.
Did person A, in their subjective experience, walk into and out of A and then die? Or did they walk into A and out of B, and keep on living?
Seems pretty clear to me it's the former.
Now move that death back just a little bit, such that it coincides with person B stepping out of point B. Does moving the death back a few seconds change things? I don't see why it should.
If I'm going wrong somewhere, feel free to jump on it.