Briairlight is an honorary warrior, since her injury makes her unfit to be a medicine cat. Sure, she can sort herbs while Jayfeather is away, but she can't go to collect herbs, go to the moonpool or the gathering, so she can't be a full time medicine cat, so what's the point of formally training her if she can't replace Jayfeather after he's gone? It's harsh, but its true, and everyone in ThunderClan is too nice to say it.
What doesn't make sense to me is Cinderpelt being reincarnated. What was the point if she was just hanging out in Cinderheart's body the whole time? She wasn't living the life of a warrior, she was watching it through someone else's eyes.
So now that the Erin's have made this bed, I am calling them out if Briarlight isn't reincarnated.