It's a bit like the idea that you "die" when you fall asleep, and the person who wakes up is a new consciousness. Or even the idea that the universe itself came into existence moments ago, and all our memories and physical evidence are a fabrication.
IIRC, Mike Okuda said in an interview once that he should've tried to establish the transporter as working through some kind of tiny wormhole or space-folding and sidestep the whole issue. The idea that you're conscious for the whole time, including being able to move around an interact with other objects while you're nominally "dematerialized" does count against the idea that it kills you. There have definitely been science fiction stories where teleportation (or brain downloading, in Caprica) is explicitly creating a copy, and the original is explicitly (and unnecessarily) killed as a bit of theater to create the illusion that something has moved or transformed.
Trek has established that the transporter though does NOT destroy the original (even ENT was explicitly blunt with that).
It certainly has the capacity to do that or make a copy, but generally only some excrutiatingly strange circumstances which disrupt the regular transporter function can get it to do that.
Or, what we could say that technically speaking the host body is destroyed but the consciousness preserved (which is then transferred into the copy body).
So, only the body is destroyed/duplicated, but the original consciousness is intact - but the dialogue indicates the body is converted to energy and simply reassembled at the destination site.
At the very least we have evidence that preserving a person's consciousness in a pattern buffer can be done for relatively short periods of time (Scotty's technique not withstanding).
What you're effectively doing is that you are preserving the neural energy (what makes a person them - their original consciousness) inside a computer... but that is excruciatinly huge amount of information as it was demonstrated in DS9 - but then again, that was a Cardassian station so, not possibly not the best example of what UFP pattern buffers can hold and for how long - because as we saw, the USS VOY crew hid a bunch of telepaths in the pattern buffer for a while from Devore inspectors - though that was hours at a time and for some it could incur some physical damage in a cumulative capacity - its possible the crew of VOY enhanced transporter pattern buffers to be able to hold people's neural energy better than what was done on DS9 - after all, they did have 7 of 9 on board who likely helped).
We live in a world where people can have artificial limbs, yet they are considered the same person they were before the accident. We even have artificial organs now like an artificial heart, yet the recipient is still the same person.
Some people belive consciousness survives death. Some have their heads frozen in the hopes of getting an entire artificial body in the future. Others look forward to the singularity and being able to upload their consciousness to a computer.
It's clear the majority, if not all people feel that what makes us a person is our conscious being, our consciousness and not our physical body. The physical body merely contains our consciousness, our being. With that being said, the idea of the transporter assembling a "clone" or recreation of your body out of the nearby atoms (as opposed to converting your atoms to energy, transferring that energy to the destination, then reassembling those same atoms back into matter) should not lead to the conclusion that you are no longer you, but a copy. If our consciousness is what makes us us, then we are still who we were.
Except, of course, the occasional emotionally split captain or the rare Lt. being duplicated and left behind.
There is a big difference between replacing biological limbs with artificial ones and creating a neural clone of a person's consciosness.
When I go to sleep, and I wake up in the morning, I do not really feel that much different.
Sure, its accurate that our bodies replace lost cells, and after a while you are not the same person you was before, but you do experience those changes over time, and there's a clear difference when consciousness is involved.
Obviously, if my consciousness ceased to exist, I would be dead. My duplicate consciousness would go on yes, but I (the original) would not wake up the next morning.
Basically it comes down to the premise that if you graft technology into the body and allow the body to rely on it and use it, your consciousness will adjust to those new sensations and your perspective will be changed as a result (in that sense, you will not perceive the world the same as before) however, this doesn't mean your original consciousness would be replaced by a duplicate consciounsess.
And, if my counsiousness is constantly being snuffed out and recreated by the body every night (which seems very unlikely), why do I still 'feel' like myself and that I'm still alive?
If my consciousness is just a copy, then this shouldn't be possible. If my stream of consciousness ceases to exist, I'm not the one waking up in the morning, and I certainly don't feel like that (and I also think that people overcomplicate this matter when they bring phylosophy into the picture - creating an overly convoluted mess and muddying up the situation unnecessarily).
Changes on a physical level do impact the person's consciousness, but to say your stream of consciounsess from yesterday ceases to exist just seems incorrect.
The more likely explanation is that the consciousness merely undergoes subtle changes - but its not the same as killing a person and making a duplicate. Sure, my duplicate would be very much alive, but I the original would be no longer experiencing things.
Even if old cells in your body die off and are replaced by new ones, that doesn't mean your consciousness is constantly 'extinguished' and being 'replaced' too every single night.
