I think what's hard to bring across is the idea that the characters don't so much need a change of heart as a change of motive and intention, just because they actually have had the heart all along. Problem with most Slytherins, actually. ^^

Ooh, sharp observation!
And to expand on it, I think you could say that perhaps the problem with the Gryffindors is a mirror of this: they have the right intentions, but they sometimes forget the heart in their efforts to see good/justice done.