First, having no memory of the past means nothing. All undead lose their memories eventually, as shown by Lucatiel and most of the merchants you talk with.
Second, there is no cannon ending for Dark Souls, as the entire premise of Dark Souls II is that the First Flame has been linked again and again and again in a constant cycle. Whether or not the Chosen Undead took the Link the Fire ending is irrelevant, because someone else would have anyway.
So no, it isn't a distinct possibility, since even if the Chosen Undead didn't sacrifice their own life to the First Flame, someone else did, and therefore would have staved off the Curse. You're also talking about a time span of hundreds, maybe thousands of years between Lordran and Drangleic. Meaning the Chosen Undead would have aged and died, assuming they weren't just killed by a Demon, Giant, or some other surviving monstrosity.