When I got the 120 I went through a similar problem an the coil was the problem. It would run great for 30-40 minutes and then just die. It was the coil and would do that when it got heat-soaked. I would guess the additional load of the heavy snow produced extra heat which soaked the coil. Some of the Deere guys have mounted the coil on the frame and not the engine to isolate it from vibration. I am considering that myself.
Heat would make sense since it's always shut down after being in use for some time.
I had a Gravely with a Kohler and key start doing something like you are describing, shutting down for no apparant reason, then being able to start. Eventually, I looked at the back of the key switch and noticed it was burnt looking. Replaced the switch and wiring harness and was good to go. So, wiring harness to switch is also a good place to look.
I've had the ignition switch out and overhauled recently and all looks okay. In fact, it was the ignition switch that was the culprit when I first bought the rig; rusted contacts was preventing her from firing up.