> Interesting that you used "brain" at the end when instead of the body as a whole, isn't it?
Not really, we know that consciousness arises from the brain, and can change if it's damaged.
> If parts of your brain started to degrade and fail, but could be replaced with machine equivalents, would the result not be "you"?
If it was replaced in such a way that the continuity is not broken, then I imagine you could slowly replace it all without necessarily impacting my experience of myself. However I don't personally believe that merely copying the information over to a new system would achieve the same, as I would be able to observe the copy go about a separate life, I would not have its experiences etc.
Perhaps you're right about that part, perhaps it could still be me without any of the current physical body, but only if that was achieved through a continuous transition, a copy that could just be extracted without affecting the original adds nothing to the original.
> That just begs the question: how much change is necessary to make "you" no longer "you"?
This question is relying on semantics for effect. You can still be a changed you without being someone else, and copy created without changing the individual is a separate thing.
> I think it makes all the difference, because the only reason you assume this continuity exists is because you have memories
Massive assumption/assertion. Memories are structural within the brain, they are not mere information, they are substance. I experience this continuity of being, and it shapes me.
> your conscious experience exists now and only now.
Everything exists now and only now, that doesn't mean that a copy of me is the same thing as me.
> Your copy will also have memories and remember being you.
Sure it will, but it will then lead a separate life, not one I will experience directly. So it is not me.
> Yet parents are interested in the continuation of their children even if they themselves still shuffle off and die. In fact, they are so deeply devoted to it that they will often sacrifice their own lives prematurely to ensure it. Why should that matter to them more than the continuation of their self?
That question is entirely orthogonal to whether or not it is the continuation of their own life. Which it's not, because they're dead and you've said as much. It could matter more to them for any number or reasons, from the biological drive to continue their line, to pure altruism. The consideration of whether a copied self would be you or not is entirely unrelated to how much people value themselves in relation to others, or why they might undertake such actions.
> self can be greater than "I".
Metaphorically, allegorically, narratively, certainly, but not in the concrete way we're talking about.
You appear to have some sort of metaphysical belief around this area, that a copy of yourself could be considered yourself in some sort of wider consideration of self that includes not only potential clones and virtual copies but even whole groups of other people. To me, that doesn't gel in the slightest - if I get copied, if I have offspring, if I sacrifice myself for whatever reason, I still die. Those others may continue, and that's great, and that might even make the sacrifice worth it in your hypothetical altruistic death scenario. But I still die in that scenario. That's not immortality for me.
I'm not trying to attach a moral value to immortality either, or say that's it's good, or more important than the survival of one's children, or altruism, or anything else. I'm literally just saying that I couldn't consider a copy to be the same thing as me, nor would children nor other people, as I would not experience their lives. Just as I am not experiencing your life right now.