I can't imagine how to deal with the problem of getting near the sheerline and finding that the strips were much closer to the sheerline at one end than at the other.

Maybe some of your frustration comes from using full length strips and/or from using bead and cove. With full length strips you need to get both ends of the "puzzle piece" right. With short strips, you only need to get one end right. I find scarfing a lot easier than trying to get both tapered ends right on a long strip.