Realistically, we're all just guessing at to to how much input Atkins has on the on-field decisions.
Based on everything I've seen and heard, and inferred from decision making... it looks like all on-field decisions are made by John Schneider. Yes, there's a pile of data given to him about hitter splits, pitcher splits, who matches up well with who based on X and Y for both hitters to opposing pitchers and our pitchers to opposing hitters and there's probably very obvious decisions based on that data more often than not.
I very much doubt Atkins pours over this data daily and questions Schneider's every move or cross-checks it with decisions he would have made with that same data, but I very much assume that large decisions that end up impacting games that maybe don't match up with the data, he's going to want an explanation, as any GM in baseball would to hold his manager accountable for decision making. I would certainly want to know why Manitply was the play in a 2-1 game.
Even looking at all the decisions JS has made, the vast majority of them work out just fine, but nobody remembers the good... the good is boring. It's expected. Mantiply in a 2-1 game though??? I struggle with that one myself but we don't have all the info JS has about his player's availability. Maybe Flu is dealing with someone nagging right now and said he was at 90% even after the off day. There's any number of different reasons that may have played into it, but sure, on the surface it looks like it doesn't make sense.
Managing is not really even a leadership position in a traditional sense. He isn't on the field playing with the 26 guys in the clubhouse, he can't lead by example on the field, or be the first guy over the wall on a walkoff HR. That's on the players. It's the players game. A managers job is mainly in-game decision making, ego management in the clubhouse, keeping them pointed in the right direction and discipline when it's needed if the player leaders haven't handled it themselves. Look at Torey Lovullo right now in Arizona... having to try and keep that clubhouse together with Marte being the world's biggest twat. Is that a failure in Lovullo's leadership because Marte is a clown?
JS is a perfectly fine manager that probably impacts the game far less than ret*rds like Aaron Boone who's only job seems to be screaming at clouds.