I'm the original owner of an 03 Z66 and we get a fair amount of snow here in southern Indiana. I've never found the 2-wheel drive to be an issue for me. The truck is my daily driver - over 300K miles on it.
At least 10 years ago I bought aftermarket wheels for it and put summer high performance tires on them. I kept the stock wheels and have been running them with snow tires every year since. Right now they've got Firestone Winterforce tires on them, which I've been very happy with. I've also run some less expensive Mastercraft tires as well on those wheels in the winter.
For years I put the winter tires/wheels on the truck around Veterans Day (it's a paid holiday for me) and put the summer tires/wheels back on the truck the first nice weekend in April. With it being warmer and milder than usual this fall I decided to push back my fall changeover to Thanksgiving weekend this year.
Keep in mind that Avalanches have a better weight balance front to back than most pickups. That together with the locking diff that came as part of the Z66 package, if you've got decent tires on the back of the truck it will perform pretty well for you. We've got 3 ways out of our neighborhood and all three require climbing a hill. I've never been not able to get the truck out when I wanted to go out, but there have been times when it's taken a few runs at the hill in front of our house to get to the top. I've never intentionally added weight in the bed of the truck but that might also be helpful but I've never felt the need to do that.