I think you've got it the wrong way round. The first part is the problem. The second part is how the problem manifests itself.
Let's take the full quote.
"No continuity of -" there wasn't a snake word for consciousness "- sself, you would go on thinking after making the horcrux, then sself with new memoriess diess and iss not resstored -"
The problem is continuity of consciousness. What Quirrell is saying is that because there is no continuity of consciousness, when you die, you die, no matter that you made a horcrux first.
I certainly don't believe that Quirrell, who has probably spent much of his life considering the problem, would be so naive as to think that destroying the original somehow gives the copy continuity of consciousness with the original.
New chapter!
This is a new thread to discuss Eliezer Yudkowsky’s Harry Potter and the Methods of Rationality and anything related to it. This thread is intended for discussing chapter 102.
There is a site dedicated to the story at hpmor.com, which is now the place to go to find the authors notes and all sorts of other goodies. AdeleneDawner has kept an archive of Author’s Notes. (This goes up to the notes for chapter 76, and is now not updating. The authors notes from chapter 77 onwards are on hpmor.com.)
Spoiler Warning: this thread is full of spoilers. With few exceptions, spoilers for MOR and canon are fair game to post, without warning or rot13. More specifically: