A project manager should never be a stakeholder. A project manager should be managing expectations and pushing back against scope creep and ridiculous demands for immediate results as part of managing the project based on available resources and the estimates of the project team compared to overall progress. They will also address situations where different interacting parts need to be timed correctly, but that would also be the same responsibility of a scrum master, because they manage the project when using agile terms.

Most places treat project managers as neutered middle men who are implementing the will of the stakeholders, which is why so many end up being the terrible type that you are stereotyping project managers to be. Those same organizations will do the same thing to the scrum master or whatever name they give to the person who is supposed to be managing the project. You know, a project manager.