Our USCG Polar Class Icebreakers work on that exact concept, and the props are actually designed to mill the ice through them.
The dynamic is called ‘dolphining’. You dolphin up onto the ice, crush it under the load of the ship, and then mill the ice through the prop.
Normally, we’d use the fuel as the weight. You get to 60 percent fuel capacity and you are too light to dolphin, let alone back and ram.
The Russians went nuclear, because every nuclear asset produces plutonium, and plutonium is worth $5,000 a gram.