The type of clip used on the original posting are called "glove snaps" and there are a lot of really bad ones out there. The largest sizes are pretty durable. They are unlikely to move sideways because they are as (Geoff pointed out) pretty heavy gauge material. The most likely falconry failure would be a back clipping scenario where the swivel or jess pulls just the right direction and slides out of the clip. The most likely mechanical failure is not the clip actually coming undone but that the lengthwise tension from end to end causes the clip to bend and then the gate is no longer a gate - it just slips free and hangs in space. On a large clip that would take a LOT of tension.

I think that the rig that David sells (and is championed by the Coulsons and various other authors) is pretty safe primarily because of redundancy. It uses 2 clips and the likelihood of both failing if they of a reasonable size for the bird being tethered (assuming that the falconer is not totally inept and checks them regularly) is infinitesimally small.

To further Scott's rock climbing analogy this would be similar to using 2 non-locking carabiners at the same time with the gates facing opposite each other.

I think our word of the day should be REDUNDANCY.