There just is not enough clearance. Plastic, fenders, brake lines, steering components etc.
I know of some folks playing with tire size and off set rims to get the room they need, and a few folks have run chains in the rear and thinner cable styles up front. I am not sure on the first gen Tundras, but on the second, I am fairly sure it even states not the chain the front. It also mentions this in the manual for the second gen and up tacoma.
Seems like a bit of a crap design, but honestly, after thrashing the work rigs in wicked deep snow for a winter, I found there was not a lot of places one could not get to in a Tundra with weight in the back and the rears chained up with a solid V bar type chain. By the time you to that spot, you were usely pushing snow past the bumper and closer to the grill. Then again, This was Utah snow, most of the time it is super fluffy and that for sure makes a difference.