I wish I could find the post but there was someone here that used a cleaner and posted the half done roof, one end looked black only broken up by the light
green of lichens, the other end was almost white. I think they just used a bucket and scrub brush with the cleaner.
Maybe someone else will recall the product? The photo was impressive.
I have seen Tilex, Bleach & water, Awsome (dollar store) Bar Keepers Friend and TSP mentioned as good choices for heavy cleaning. Also those white mr. clean erasers.
I did find this post with pictures that may give you some encouragement.