I get why people prefer her to die, simply so it makes it a more 'edgy' story, but honestly that is a poor move. Because realistically with how close those 2 are you would earn his permanent loyalty saving her, and when she dies he would be a broken man for quite a while. Now I know the story basically decided to limit him to bawling until they burn her then it's just "bummer". It would also have strengthened her body giving them another physically capable person (if not martially inclined) I just find it poor story telling, and I always see people accusing Japanese Protags of doing the "I gotta hide my power" routine.
Now it could be a translation issue, where he actually didn't have anymore and couldn't get more in time, which would be acceptable. But it seemed more like he made the conscious decision not to save her.