From my experience, chins & pulldowns are equally good if done properly-touching the lower chest and using a good amount of resistance. Try close grip pulldowns using a V-bar, as well as trying medium hand spacing with palms facing together & palms facing out grips.
You don't have to do more than a few good sets of pulldowns or chins. Instead, add some DB/pulley pullovers, one of the best lat exercises.
Hard to tell but it looks like chins/pulldowns are for width and rows for the middle of the back, but both should be done anyway to cover all bases. On rows, try close-grips using a V-bar.