We live in a world of subtly and nuance. It's rare that there's a simple 'black or white' answer to a question.
IMO compute power (GPUs) played the bigger role. DeepMind themselves say that training AlphaGo wouldn't have been possible without access to Google's large-scale infrastructure. They've been training it on thousands of machines simultaneously. That's not to be confused with the hardware necessary to play the game, which isn't much and can be done on a single machine. Only the training phase has these extreme hardware requirements.