Consultants have as much power as the studio or publisher gives them. When Nintendo hires a consultant to act as the team's executive producer/director (e.g. Masahiro Sakurai via Sora Ltd.), then that consultant will have as much power as a director of the project, granting him broad authority to do what he wants. When the Tony Hawk Shred team hired a physics professor to consult on the physics of skateboarding, they tasked him with coming up with a control scheme that felt like actual skateboarding. He didn't have any more power or responsibility than that. In my experience, companies hire consultants to ask them questions about their specific area of expertise and then they can choose whether they want to listen.
I suspect your question is primarily about a particular consulting company about Diversity, Equity, and Inclusion. These folks aren't any different from the WWII consultants that were hired by the Call of Duty team to make sure the games felt real. Consultants are usually there to provide advice and answer our questions. It is up to the company's leadership above them to decide whether to make the choice. A company like Sweet Baby Inc. is generally brought on to answer questions about the representation of people within the game they're consulting on, since that is their field of expertise. Whether the devs doing the narrative design, quest design, cinematic design, character design, item design, etc. take their suggestions is dependent on the decisions of the team leadership.
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