Limited slip's limitations.
What you're expecting is a locker, which has teeth/splines that engage the carrier itself so that both axle shafts spin the same speed as the carrier no matter what kind of traction the wheels may (or may not) have. This is the only way to get a wheel to spin with the other in the air.
Clutch type limited slips work by the natural way gear teeth want to force themselves apart from each other. This compresses the clutches against the special plates that are keyed to the carrier (those little tab parts you'll see in there). This kinda helps transmit some extra traction to the slipping tire, but only if the other tire has traction because, "bias ratio". All limited slips have it, it's a factor of how much traction the slipping wheel has. Say a L/S diff has a 3.5 bias ratio (common Torsen ratio), that means the carrier can transfer 3.5X the traction generated by the slipping wheel to the wheel that still has good traction. Obviously if your wheel is in the air, 0 X 3.5 = 0. Also very little traction, still means very little power transfer. A Torsen L/S does much the same thing but with gears, but the principle works the same.