I don’t think it was a question of “who trains him” so much as it was a question of “do we have him trained or not.”
I feel like she puts emphasis on Luke, they probably thought that Luke had the best chance of turning him away from the dark side. I don’t think training in itself would do that. I don’t know. I’m not sure if it matters though. I think the information in that line (whatever it implies) isn’t needed. And there being “Vader” in him is obvious anyway, Kylo is enough of a Vader Junior without it being spelled out.