Lily comes across as an obsessive teenager who is not self-confident, or logical, and might even be a little mentally ill. A teenager wouldn't be obsessed with a mother she never knew - she might wish to have "A" mother, and could make up an image of what her perfect mother would be like. But the fact that the family doesn't talk about her mother really wouldn't weigh on her mind since she never knew her. And only an obsessive kid would think up a scenario that includes believing she has to die if she caused her mother's death - that's bordering on severe mental illness.
Now, if she was a normal, "motherless" kid growing up in a family comprised of not-quite normal relatives, and she then stumbles upon some facts or evidence that leads her to consider that her mother's death wasn't an accident, that's believable. She would want the truth to come out, so her mother could be avenged. But to think she might have to die because she might have caused her mother's death just by being born is unsettling and uncomfortable.