You just blew my mind (no pun intended).
In this thread, we're basically alluding to the Hard Problem of Consciousness[0]. Consciousness is hard to define, and even harder to prove, and so conversations that talk about it become difficult.
There's certainly a certain subset of the brain you can take out and still have no effect on consciousness. Assuming you could replace small parts at a time, what would happen? At what point does the "mind", "self", or "consciousness" get lost?
[0] https://en.wikipedia.org/wiki/Hard_problem_of_consciousness
If we don't even know what consciousness is, how do you measure the effect on consciousness of removing arbitrary amounts of brain matter?
Damage a brain enough and brain death occurs. By the same token, after a certain number of neurons are lost, there is no self. Whatever consciousness is, it can’t be sustained with a sufficiently compromised brain.
This is observable in late stage dementia patients, it gradually becomes difficult to think and in the final dire stages their sense of self degrades too.
At best, you can be an external observer. You can't truly understand what the person is experiencing.
A common thread I see among people with brain damage is that they don't recognize it. For example, with patients with hemispatial neglect [0], if you ask them to draw a clock, and then only draw the right-hand side of one with the numbers 12 through 6 and they don't draw 7-11, then try to ask them where the 9 is, they might insist it's there. If you ask them to point at the 9, they'll either point to the wrong number or not point at all. And if you tell them they're not pointing or are pointing to the wrong number, they'll insist they're right.
But even with all that knowledge, it's still hard to imagine what that person is actually experiencing.
So called 'terminal lucidity' exists sometimes.
One theory is that is enabled by the immune systems shutdown, so inflammation ceases to interfere for a short time.
The problem with that is it still fails to explain the few cases, where shortly post-mortem it is found that there is not much healthy/able to function somewhat normally brain left to speak of.
Rather 'spooky'.
I think it doesn't "shut down", maybe fades away to a different, low-power mode. When you go to sleep and then wake up, here is still a continuity, because of the same underlying structures and processes from which the consciousness emerges. So it is like a 4D thing.
That continuity never really breaks (until death? which is like singularity) and I think this is what makes you "you". You can't copy/teleport it (kinda by definition), but you can extend/transform it.
Perhaps a "Ship of Theseus"-style approach would work — gradually replacing neurons one by one, or brain sections piece by piece. Alternatively, the brain could be extended with peripheral "devices" connected to it with bandwidth comparable to the interconnections within the brain itself, up to the point until the biological part becomes irrelevant. This is similar to how a portion of the living brain can be removed while still preserving consciousness — as neurosurgeons often demonstrate.