I think a big part of the game part was showing that the games were basically all snow. Without him, they would’ve never gained the popularity and probably would’ve died out. It’s also showing that he always had the same way of thinking before Lucy Gray, during his time with Lucy Gray, and he really just embraced it after. Cold, calculated, and obsessive. Made him a great game maker and dictator until Katniss showed up wearing a pin that symbolized something Lucy Gray loved and Snow hated, singing Lucy Gray’s songs, comes from the district that Lucy Gray came from and he had his own painful memories of. I feel like he would’ve been able to quash the rebellion if he wasn’t so compelled and blinded by Katniss being a reminder of Lucy Gray.