Oh that's such a good theory, especially since they call out their "first" meeting at a ball where they supposedly never spoke? Could actually be the 1st timeline of how they actually met.
In fact, I think you're right but there is a first master of the cursed sword...and I think it's him. So, in the second life, when the sword come to Eki, he feels big remorse because it was supposed to be his curse and it explain why he don't want to kill her and wait she regain her body. Because he know how it work and how she feel since he lived it himself. And the sword wont say it, it's not in it's interest she discover all of that.
I think there is love in all that, perhaps in the first timeline events, she could have help him. Perhaps there are also deep understanding of the pains caused by the cursed sword and the plots and also culpability of letting the other suffer from it without knowing the biggest picture.
But everything I wrote is pure theory, I didn't read the novel.
theory that yurien turned back time before echi turned back time. so there are three timelines: the first (when i'm guessing he fell in love with her but she eventually turns into the demon sword, and she doesn't have memories of it because she was not master lvl yet), the second (she still turns into the demon sword and he tries to save her, he can't bring himself to kill her), and now the third which is the present