The reason on-ice shot differential is so important to evaluating hockey is not because it is an end-all and be-all statistic. The reason shot differential carries so much influence is because it’s the best (publicly available) statistic that predicts future goal scoring.
Expected goals arose as a way to better describe the quality of the shots teams take. Expected goals take into account how close to the net a player was when he took a shot, whether the shot came on a rush or was a rebound shot. Expected goals isn’t a perfect number, either, but it can tell us a whole lot about the quality of chances a team generates.