No truth to this. The locking mechanism is a simple sliding collar coupling the two shafts together. The only thing required is that the ring gear may need to rotate slightly in order for the locking splines to line up so the collar can engage, but that's no different than needing to engage the clutch slightly sometimes to get your manual transmission to go into gear when stopped in neutral. No wheel rotation is necessary.

Any traction device, including an LSD, in the rear end will be prone to fishtailing on ice. That's because a locker or LSD will not allow one wheel to spin freely while the other one continues to maintain a grip, however tenuous, on the road. It's the non-spinning wheel that provides the lateral traction that controls fishtailing. By forcing both wheels to spin at more or less the same rate, the locker/LSD loses lateral traction and slides sideways quite enthusiastically on slick surfaces.

The open diff is unsurpassed in that area, since it only takes a fraction of a percent difference in traction to transfer all of the wheel speed to the slipping side, so there's usually one wheel that doesn't spin. The open diff is generally significantly superior on dry or uniformly slick roads, but finds itself at a significant disadvantage on surfaces where each wheel has widely different traction.