Not quite. You’re describing our brains as a ship of Theseus, which is fairly accurate. But our consciousness is always on while alive. Even asleep and in near-death or temporarily dead our brains don’t fully stop or die. Though our brains don’t actually replace neurons quite like they replace all other cells. When neurons are damaged, those pathways are lost. Our brain is redundant enough that rarely manifests as a total loss of ability. And when it does, our brains can eventually route new pathways. If enough of these are damaged at once, it can totally change a person’s personality.

But transporters turn matter into energy, those patterns are transmitted elsewhere, and energy (or different energy if stored in a pattern buffer) is reassembled very much like replicators. In this case the entire brain and body is stopped, destroyed and re-created. This is, for all intents and purposes, death and cloning. People have trouble with this because to anyone NOT transported, it looks identical. But the person absolutely stopped being alive and a new one was borne that thinks it has always existed.

And Star Trek backs it up. The classic transporter accident that makes a clone of someone? If the transported person is still the same consciousness, what is the clone? Clearly that person isn’t controlling 2 bodies with 1 consciousness. Which is the “real” McCoy? The answer is whichever wasn’t disintegrated, or neither if they both were as part of the transporter process.