Can't speak to the general grabbers, but have dealt with both KO2s and AT3Ws.
Compared to a KO2, the Falken gives up some deep sidewall protection. There is about a 4 inch band coming in from the tread that is as thick as the KO2, but it doesn't run all the way to the center. If you regularly hit your rims on rocks, you might notice this. However, I've never had a problem with it on desert/mild to moderate rock trails.
Snow performance of the AT3W however is significantly better. Its a nice deep tread, and handles that well. Not the stickiest snow AT out there (that goes to Cooper ATWs which aren't made anymore), but more than good enough for most use cases.
Basically, you give up a little sidewall to get better snow performance and a lighter weight tire. IMO, the KO2 is still the best rock oriented AT out there. However, the Falken beats it for mixed driving and snow/rain.