To be fair, the wizard must have told them that her cause was hopeless. Not only was her wound caused by the magical poison of Thulsa Doom, her life was already forfeit due to the demonic pact she made as part of the spell to save Conan's life.
Anyway, Conan is a killer, not a doctor!

I've updated this write-up with the details about the pyre not supposed to be able to burn, and Conan wearing her gem as she burns.