A manager needs to make sure the parts are moving. That there is a process and a plan. Make sure a player that is struggling gets the help they need, a hot player stays focused and to make in game personnel moves. When to go to the mound, take the heat and get their player off the field if there is a bad call.
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Making sure the parts are moving is a vague term.
Having a process and a plan: Simple stuff in baseball. Lineup, starting pitchers, bullpen, whatever. Despite all the fan complaints, differences between manager moves are minimal as to won-loss record over the course of a year.
Make sure a player that is struggling gets the help they need: Players struggle on every team, every year, all the time. Every team has assistants to help. Yet struggles, slumps and hot streaks continue every year. Remember what a great pitching coach Leo Mazzoni supposedly was with the Braves. Suddenly he wasn't so helpful with the Orioles. I wonder why.
Make personnel moves (in game): See process and plan.
When to go to the mound: See process and plan, most of which is set now by the front office.
Take the heat and get their player off the field if there is a bad call: Totally irrlevant. But it would be fun to see Early Weaver or Wally Backman argue with umps.
This stuff is just lowering the microscope to over-analyze mostly inconsequential details while missing the big picture, which is collecting talent in your system and in the major leagues. And that's the front office's job, and that's where the important stuff happens.