As others have mentioned, when it comes to snow, it depends on the conditions as to whether they are any good or not.
A thin, powdery coat is going to be different from a thin, wet coat. The same goes for deeper snow. So the ability to drive in the stuff isn't necessarily the issue but the type of coating and how much.
A mud tire can handle deep snow well enough and modern designs have changed mud tire tread the design. So they are a lot better than they were.
Snow tires, and all season and all terrains to a lesser degree, are designed to trap snow in the tread since nothing grips snow like more snow. Siping plays a part as well. Snow tires have a lot of them. Mud tires tend to have some but not a lot. Again, all season and all terrains are in between.
When it comes to all season tires, there are some that are more of a rain tire than an all season design. You can tell by the tread design. They usually have large water grooves and not much in the way of a tread pattern to grab and grip loose stuff.
No tire is going to be THE answer for everything. They are going to do some things very well and the others are going to be some kind of compromise. A mud tire is designed to excell in mud. A snow tire is going to excell in snow and ice to some degree. The rest is going to be a compromise with a focus towards certain criteria. Like all season tires, some ATs are going to be pretty good in snow and others are going to suck.
Here in PA, we often get thin layers of snow and I'm often out on the roads before the snow plows are. "Good enough" isn't good enough for me. They have to be excellent and a mud tire isn't going to be excellent. There are times that the only reason I made from point A to point B with out a wreck or ended up in the median was because I didn't go " good enough", especially the few times we had flash freezes.
So, if
@00t444e wants to run mud tires and they are "good enough" for him, more power to him. We aren't here to prove how much better we are than others or to compare "equipment" sizes.