The thing with the games industry, is that there are no real agreed upon, exact definitions when it comes to job roles.
For example, one dev might have a team of 3-4 producers working on separate parts if the game, all reporting to a lead producer, who in turn reports to the studio lead.
In this case, the role of producer has about as much responsibility as any other role, in that the studio lead has more control over the project.
Production staff in this case just function as people who are responsible for making sure that schedules are met, and features are completed.
In another example, you might get an external producer, who does little to nothing in terms of the development of the game, other than swan into the studio once every 3 months and ask why the game isn't finished yet.
When you're talking about AAA blockbusters these days, the people with the most say in the game are usually at boardroom level, anyway.
And much of what they say is usually down to meeting release date windows and ticking boxes, anyway.