You said it much more eloquently than I did, but your post was what I was trying to say.
To carry a bit further (I'm nothing if not verbose), we need to remember that CM is not an activity. It is a discipline, a specialized form of management discipline.
Folks, please don't confuse the definitions of management as a discipline and management as a group of folks who "run" the organization. Unfortunately, because of confusions about meanings, many often assign the first idea that comes to mind. They grab the low-hanging fruit rather than investigate to find the real meaning. Or, they simply accept what was stated because they don't know any better.
For example, a few years ago no one ever thought of having a CM specialist doing a software build. Then someone discovered that you didn't necessarily need SW expertise to perform the build function (probably a vendor who likely proposed saving the expense of hiring a build engineer "because with our tool, the CM person can do it").
However that happened, we started seeing job posting for a CM Specialist with a job description that included "Must be experienced in performing builds."
As to the question of "management versus production/development," CM is in my opinion most definitely in the management category. Please note that no document that purports to be a "standard" allocates actual activities to a CM specialist. All the "standards" I have seen, not all of them I'm sure but still a large number, state the activities that must be done, but not one tells us "how" or "who," except the nebulous reference to a "decision makeing authority or body." A couple of those standards even go so far as to add "(sometimes called a CCB").
With that said, CM is in my opinion very closely aligned with another specialized discipline called "industrial management." Although the products are different, their relationships to the more generic management discipline are similar. The easiest way for me to express it is to say that on a project, project management is concerned with a lot of "stuff" like personnel, financials, schedules, human resources factors, customer coordination, product delivery and myriad other things. To accomplish all that, the project manager may have a staff of people to oversee/manage specific pieces (recruiting manager, project finance manager, etc.), each of whom applies the principles of his/her specialty to satisfy the projects management requirements.
In our case, we are, or should be, concerned with managing the product. (This is another place where terminology is often cavalierly tossed about, i.e., product vs project. As in, "Our project is the xyz application." No, the "project" is to CREATE the xyz application; thus, xyz is the product.) Ensuring that the product goes through all the right loops to guarantee(?) that it is the correct product is the domain of the CM discipline. It is the CM specialist who does all the planning, organizing, coordinating and controlling the CM processes and activities. (Control here refers to taking feedback and making adjustments in the CM system.)
In a nutshell, CM is definitely in the management arena and not a creating activity. On the other hand, "build" is a creating activity. Therefore, "build" is not part of CM.