...Somehow they don't look anything like as secure as attaching the ground wire with a screw.
That's why I don't like to use 'em. Not only are they a pain, but I've seen them installed incorrectly several times (which led to a really poor connection) and even when done properly, some of them didn't seem to grip the conductor very well. I prefer a ground screw any day of the week.
That being said, if I ever ran into a job with a bunch of old boxes that didn't have tapped holes for ground screws, I'd probably use the clips instead of spending all that time drilling and tapping.