Yeah, but that wasn’t a time travel episode. I seem to recall reading that the leap was due to something done during the City on the Edge of Forever episode…
Well, by getting USA into the war and having them develop nuclear technology, the stage was set for the space race and the discovery of warp and other things. I think that was implied in CotEoF, and maybe explored in depth in one of the non-caanan* novels.
*The Promised Land, Tech Utopia. 
First, I already answered that: it depends on which of the characters you ask, though for most of them, most of the time, no death is involved. Second, I was responding to a post that stated unequivocally that you die. Third, in the OP you presented a false dichotomy, and dealing with that does open the door.
Rather than ask Mr. Roddenberry, I’ll stick to what his characters say. Most think they’re still themselves. Some (at times) wonder if there is a life-and-death issue. Which is exactly what would be the case for any real transporter, no matter how it worked.
Well, depending on your (universal your, not you miller personally) religious beliefs, there’s the soul. Even if a transporter could do all that stuff to a house, you pop a person in and what happens to his soul? Are transporters merely creating millions of soulless shells in the place of humanity? Probably not, but you never know.
Okay, so when in canon (not books) is it ever presented as destroying the original? I can’t recall it. McCoy’s aversion was just that he didn’t like his “molecules spread around the universe.”
Anyways, the OP’s question is a bit more specific: is it the same atoms that are reassembled. Most of the time, it seems the answer is yes. Not only do we have McCoy’s lines, but they specifically talk about a “matter stream” that contains everything. Other times, they talk about energy masses that contain the original matter. But I only recall two times where the atoms are clearly reassembled from another source.
The first one has already been mentioned. It’s “Second Chances,” the TNG episode where Riker is duplicated due to an unusual and dangerous way of using the transporter. The Riker on the ship was the result of a second scan while he was still in transport, and the Riker on the surface was the result of the scan bouncing back and rematerializing on the surface. So only the one on the planet can possibly contain the original atoms.
The second is the DS9 episode “Our Man Bashir,” where an incomplete transport results in the data from the transportees being stored in the station’s computer. The data for the crew’s brains are stored in computer memory, while the data for their physical form is stored in the Holosuites. As holo-matter is explicitly not the same thing as real matter, it can’t be the real matter being stored. And, as they use a different transporter altogether to fix the problem, it seems they must’ve been reassembled out of something else.
At least, the episode makes no mention of them physically moving the matter from one transporter to another, nor of using the working transporter to transport it. It’s all about getting the station’s computer and Holosuites to interface with the transporter. Well, that and keeping the holo program from ending and deleting the physical patterns.
And Quantum* Flux Capacitors*!
What do you do when there is a yellow alert?
Slow down.
… What … do … you … do …
What would happen if I removed a tiny crumb of skin from your fingertip, and replaced it with an exact duplicate? Would you still be you, or would you have died? What if I did that hundreds, thousands, millions of times?
Thing is, this is exactly what is happening in your body every day. After 10 years of life there isn’t much of the original you left, you’ve replaced almost all the atoms in your body. In a very real sense the person that you used to be doesn’t exist any more and has been completely replaced by a new person. If you don’t think that you gradually die and are gradually replaced by a copy every 10 years, then why would you think a transporter would kill you and replace you with a copy?
The thing is, our selves are not collections of particular atoms. Atoms can and are added and removed from our bodies every breath you take, every drop of sweat you shed, every drink of water you take, every bite of food you eat, every time you eliminate waste, every skin and dandruff flake that falls from your body. The important part isn’t the exact atoms, the important part is the pattern of those atoms. And even then the pattern can be modified and you’re still you–you don’t look the same today as you did 20 years ago. If you lose a finger you’re still you.
As was pointed out above, a difference that makes no difference is no difference. There is no such thing as a “soul”, except the product of our minds, and our minds are the product of our bodies. The idea that a person without a soul could walk around and you wouldn’t be able to tell the difference is proof that there is no such thing. Or, if there was such a thing there would have to be some way to tell the difference between a person with no soul and a person with a soul.
I mean, how do you know that other people are conscious and have souls just like you? Maybe they just pretend to be conscious, and act as if they have a soul. Except I contend that you can’t pretend to be conscious unless you actually are conscious. Or another way of putting it, you have to be aware of your internal states and able to model the internal states of others to simulate consciousness, because that’s what consciousness IS.
This debate about whether a copy is the same as the original makes some sort of sense when we’re talking about physical objects. But the obvious analogy is not an object but information. If I take a picture with my digital camera I create a data file on the camera. If I copy the file to my computer is that file the original file? Or a copy? But if the copy and the original are exactly the same what makes one the copy and one the original? It doesn’t make sense to speak of copies, since what is important is the string of ones and zeros, not what part of a computer memory those ones and zeros are stored on.
Of course, canonically people in the Star Trek universe consider transporters to move you from one place to another. Yes, they are aware that weird things can sometimes happen and people get duplicated or modified by transportation, but they just shrug their shoulders and move on. Some people think transporters are more sinister, but those people are regarded as cranks.
This doesn’t mean that the people who think transporters move you are correct, just that nobody who uses a transporter worries about it. It seems pretty clear that if you refuse to ever use a transporter then your career in Starfleet would be over.
“The Enemy Within” from early in the first season of TOS has Kirk being duplicated by a transporter malfunction. That argues against transportation being just a stream of the original molecules, but there is plenty of evidence for this also. It makes sense, since where else would the matter for the transported thing come from - especially at the remote end, where there are no pattern buffers.
Not completely duplicated, tho. His soul, emotions, essence, whatever you want to call it was split between the two bodies.
One body was amoral, maybe even immoral, and could make command decisions. The other body was kind, caring, good, and so meek he couldn’t choose between a soda can or bottle.
What does THAT tell us?
:dubious: <— Trek smilie
The question cannot be definitively answered because the canon on the matter is self-contradictory.
As a result of a transporter accident …
And quantum time discrepancies caused by the Temporal Wars.
Actually, we could just blame Wesley.