Riding in deep snow is not so much the problem. The problem is what is under the snow? If you know the area and are confident that there are no hidden obsticles, then the ride across a field of fluffy white snow is no big deal.
My challenges usually come from blow down logs, rabbit holes, loose rocks that roll when stepped on, traction on mountain sides covered in snow etc.
The other concern is crusted snow. If the crust is thick, it can be fairly abrasive, especially on the horses shins as they break a trail through the snow. We also run intot he problem of riding on South and West facing hillsides where the snow is not so deep and then finding a North or East face that doubles the depth of the snow. The various creeks and streams that I love to practice crossing in the warmer months, become sheets of ice.
We ride all winter, I just choose more carefully where we will ride. Often choosing South facing hillside where the snow is not as deep and we have more sun to warm us. I choose areas that are more sagebrush or open country and avoid the heavy timber that can hide tangles of branches and where the sun doesn't reach to melt the snow.
This west facing hillside is almost melted off from the storm the day before
I'm a whole lot more willing to ride across an area that I knew was a hay field in the summer than to cross an are that I know is talus and broken rocks,
And often times I can follow the trails the deer and other wild life make.