Unfortunately...people nowadays are just plain stupid and drive way too fast in their traction controlled street treaded SUV. Too bad you are unable to chain up because you can pretty much do whatever you want with a good set of chains except run really fast down the road. Sounds like your driving would be best on a set of picked snow tires. Of course that doesn't help you during the summer and requires two sets of tires.
I have to say that out of the many tires that I've run on 4wd trucks, the next step up from the classic AT tread style was a Goodyear MTR. I had a set of 37x12.5's on my Yota crawler and those ran awesome through deep snow. I would haul ass without chaining up and they did great. Of course ice is a whole other ballgame but they did work well in snow. The old Swamper Trxus MT worked pretty good too for a fairly aggressive mud tread.
As far as diffs go in snow/ice, I've had LSD's, Detroits, Spools, and open with all different sets of tires over the years. My preference is always towards the locking diffs because I know when I hit the throttle both tires are going to spin on that axle. Having said that, you can't be dummy behind the wheel. Turns require finesse and slippery conditions in a straight line can all of a sudden turn into some underwear changing sideways action. Gentle with the throttle and paying attention with the steering wheel feedback will get you through the thick stuff with a traction differential. I'll take a spool over an open rear diff any day of the week.