It wasn't the song that changed their minds. Shinra explains what happened afterward: he was using a sphere-based system to amplify the music, and that also amplified the energy from the Songstress sphere, which is why Lenne's image appeared. It also amplified and broadcast Lenne's emotions, as recorded in the sphere - her sense of loss and futility, the hopelessness of war - which is what affected the audience.Also, I don't see what's so cheesy and farfetched about 1000 Words? Is it the fact that a four minute song (+graphical aids) managed to get two groups on the verge of war to become friends?
As for Leblanc looking like Yuna... she was clearly using something other than *just* Yuna's grid. Some kind of magical disguise, a simple illusory trick.