Without seeing the conversation, we can't know if Hajime's parents really do see her as the "spare daughter" like that, but the important thing is that it's how Hajime perceives her role in her family. Doesn't matter what her parents feel, if the impression they give her is that they don't care what she does.
Hajime's story is definitely one of tragedy, though. To lose someone so precious to her is terrible, but to also feel like she can't escape Kyou's shadow even in her death, compounds that feeling that she's not able to live her own life. Add on her love for Momoka that she feels she can never express in the wake of her sister's passing--and Momo's clear lingering attachment for her--and Hajime is sort of chained in place, unable to choose her own way, her own happiness, because how can she be truly certain she loves the drums, this band, for her sake? And not because it's her sister's plan, and the mirage of Momoka's proximity?
The letter is a bombshell, though. Only knowing the last lines and who the letter is for, we can't say much about the deeper implications, as that would rely on Kyou's reasoning and context that is hopefully in the body of the letter.
I do see Hajime sitting on it though, in the face of Momo's clear love for Kyou, not wanting to taint her memory with this letter from beyond the Veil. The stress of it might affect her, and she'll let slip its existence in a moment of weakness--possibly to Himari or to Shiho instead, who will get involved at that point?
No real clue from here, but I trust the author to handle it well and ensure Hajime and Momoka come out the other end okay.
Thank you for the TL as always.