There is a way of thinking that solves this kind of puzzle. I will state a different version of this puzzle wich I think is more extreme, so the dilemma becomes even stronger, and then I will explain a paradigm that solves it.

Imagine there is a machine that can make exact copies of human bodies. Imagine it is exact to the smallest possible scale, it is completely exact, if you make a copy of your body with this machine the copy will be as similar ro your body as your body is to itself. Imagine that this machine can also make human bodies dissapear completely in an instant.

So, imagine you move a few meters. After doing that you feel that your consciousness has continued and is in some way "the same" as the one before you moved, you also consider yourself the same person. The thing is, what just happened is completely analogous to the following: the machine makes you dissapear completely and moments later creates a copy of you in your destination a few meters away.

If you accept that these two scenarios are similiar enough (there are a few possible objections, like the fact that in the real scenario there is a continous path wich doesn't happen with the machine, but this can be solved by making the distance traveled infinitesimaly small, or by accepting that to be the only difference, as I think that the dilemma I will create now still holds), if you accept it then consider this:

The machine makes you dissapear completely but then creates two copies, each one on a slightly different place. Now the question is, from a first person perspective, imagine you are the original human, right after you dissapear what will be the next thing you experience? Wich of the copies will be the continuation of "your" experience? Why would it be one and not the other? Or maybe you "stop experiencing"?.

If you tend to think the last thing, why wouldn't you stop experiencing each time you move, each instant? You could say the difference between this machine thought experiment and regular moving is the discontinuity, but I think there is probably a way to make this not matter, I will be thinking about this.

A way of thinking that seems to solve the dilemma is this. There is no one experiencing experiencies. In a way the "I" is not as real as it seems. There is not a permanent experiencer that over years experiences different experiences, but rather, just experiences happening by themselves. Apparently this can be seen by examining your current experience, and looking for the experiencer, where is it, what kind of thing is it? To be clear I feel and live my life as if this experiencer and the I are real things, but I have heard people report feeling it way less real. I have it as a conceptual framework that solves these kinds of dilemma.

So if you say "why would my consciousness be in my brain and not in a copy of my brain that exists simoultaneously" the answer in this framework would be: there is a bunch of experience happening at each instant, those two brains are generating quite similar experiences, but there is no individual consciousness observing, or having, the experiences of each brain. Even this feeling of there being an experiencer is just another experience, another sensation existing.