Julia and Leagwen, thanks for the feedback. I didn't sieve the powdered sugar before I made the icing. I just whisked it with the cream of tartar. Maybe that's the culprit. Do you think I could get better results using pastry bags with tips?
I pipe with cones all of the time, and vastly prefer them due to the control and precision (finer lines than tips) they afford - and the fact that they don't require clean-up. It may be you just need to practice with the cones more, but by all means, try metal tips and bags, and see what you think. (P.S. I never sieve my powdered sugar, but we typically get a very fine - 10X vs. 6X - powdered sugar here in the US that rarely, if ever, plugs cones or tips. If you are elsewhere in the world, though, some powdered sugar could be quite gritty - but this will affect your ability to pipe easily out of both cones and fine metal tips.)