Because whatever you are, you're not a single neuron, you're the process that is happening across your entire brain, or at least a large subset of it. If that's true then as long as that process is maintained, there's no way that replacing individual neurons can replace you with a copy.
I wouldn't step into a transporter, but I don't see a problem with replacement spread out over time. One interesting things about this scenario is that it can't lead to multiple copies, which makes sense if the final thing is the original and not a copy.
So what's the role of time here?
I have the same instinct that replacing some arbitrary small percentage of my physical brain bit by bit seems to preserve that idea of the self as a continuous pattern. In the same way that we replace the cells of our body gradually.
But something about the role of time here seems a bit counterintuitive. Is there a pace of replacement which would break this continuity?
And the role of the units replaced seem arbitrary also. We can both agree that replacing one neuron preserves you, and replacing the whole brain in one go does not. What percentage of the brain can be replaced in a single action to preserve self, and why that particular amount? It's a Sorites paradox situation. If we set a limit, then we need a good reason to say one neuron more than that is too much. And if we can't say that and there's no good reason to draw a line at one nueron more, than why not one neuron more than that?
When we draw such lines in naming conventions (Like how many grains of sand make a "heap") or in laws (Exactly what age should be the age of consent) we know we're being at least a little arbitrary for practical reasons. But if the question is "Where is the line that makes you truly yourself in terms of brain replacement" then the line isn't just a practical arbitrary tool, it's saying something important about identity. Is there a gradient? Starting at X percentage of your brain replaced you're a certain percentage less "you" scaling up as you take larger chunks? That seems counterintuitive because whether your experience is a continuous pattern should be a binary fact, no? You can't be sorta kinda continuous, or sorta kinda yourself. We don't even have any tools to possibly measure that.
Because, as a limit, the end result is physically identical to just making an artificial copy and plopping it in your skull. I think we mostly agree that doing that would not preserve the continuous pattern of "you". So it's something about this intermediary stage that's supposed to preserve "you".
What if, instead of days or months between operations, the procedure was enacted by some sort of nanobots. Is there some period of time that might serve as a hard line? If your whole brain switches over in the time it takes to snap, has continuity been preserved?
I can feel the time objection may seem less strong that the size of replacement at a time issue, but let me add a wrinkle.
What if, those clever nanobots, instead of destroying the old brain cells, carried them away from your body and reassembled them in their original configuration. Maybe within a robot body. Now we have your original organic brain and this new artificial brain, both of them with all your memories and feelings, each thinking it's you. Are they both you now? Is the original reassembled organic brain less you because there was a microsecond gap between it being dissassembled and put back together? You seemed pretty comfortable saying that the artificial brain replaced part by part was you for any meaningful purpose. Does that mean your real brain is now the "copy" and it's recollections and experience are not continuous from yours? Which body are "you" in?
See, replacement doesn't actually free us from the copy issue. In fact, the whole transporter thought experiment comes from the Ship of Theseus which started as a question about replacement part by part and even included the question of what if the original parts were reassembled.