Well from what Asuna said didn't Kirito want the sword for himself? There's durability on all the equipments so it's probably a good idea to have some spares and having one that's as good as the one he usually uses is probably preferred. And he did say he wanted a sword that's as good or better than the one he had and it's pretty close to the same stats. Also it said Dark Repulsor + 0 so I'm guessing you can enchant/enhance it while you can't do it with Elucidator since there weren't any + number on it.