"Too much influence" is a complete independent problem and I'm going to do my best to make the most sense out of this post that I can. Managers should be in control of what happens during games and there are a bevy of sabermetric ideas that can be employed from lineup construction, to matchups, to shifting, to the running game, and anything else the manager normally calls the shots on. Sabermetric thinking isn't just referencing new fangled stats (that's part), it's really using any empirical evidence to make decisions rather than eye tests and gut feelings.