I admit I could be wrong, but I've never seen gold and other finches eat greens for greens sake. Usually there are some kind of seeds involves.
I suspect that your sunflowers are infested with sucking insects that congregate under the leaves. In my garden, most typically, they are lace bugs and tree hoppers, sometimes leafhoppers also. See if there are ants marching up and down the sunflowers -- they are useful guides when used for finding them.
In my garden the finches are currently eating lyreleaf sage seeds that have gone to seed (I hope they eat most of them so the sage doesn't spread too quickly). A little while ago, they were taking apart chickweed seed pods and scattering beet, mustard and kale seeds from the plants I allowed to go to seed.