Can't speak for how it works in the new CSO squadron, but if it's anything like the rest of the AF, it works something like this....
First off, he's the "Operations Supervisor"; it's a duty officer position, except it's not just some faceless position that an O-1 fills because there has to be a warm body there. Instead, it's the Squadron Commander's direct operational representative on a daily basis -- he's the dude who is running the flying operations. In the fighter world, we called the position the "Top 3", because the position was filled by the top 3 positions in the squadron (SQ/CC, SQ/DO, and SQ/ADO).
It's meant to be the senior operations guy in the squadron, and it's his job to make sure the scheduled flying is executed safely, efficiently, legally, etc. He's the interface with maintenance, matching up scheduled sortie lines with the flight-suits that will fill those lines. He's setting the flying status based on weather (and working with the SOF, natch). He's making sure the IPs have their own act together and aren't going to plan/execute that is stupid or not legal (you might be surprised how much this happens given the volume of flying that takes place in the training environment).
Yes, he has all the authority in the world to turn crews back and say "you're not ready to go fly...you're done for the day".
He's not just a face that stands by the door and gives some briefing.