Lyrics are tough to translate, because of the license artists take in playing with the language they use, and the myriad of ways you could interpret the words. One thing I noticed:
最後に残るものは小さくて脆いもの
それを希望と名付けこの胸に埋め込んだ
you translated as:
What lingers in the final moments is small and fragile
That desire was named and buried in this chest
I think what "それ" refers to in that second line is the 小さくて脆いもの, so it would be closer to:
The small, fragile thing remaining/lingering in the end
was named hope and buried in this/my chest.