Specifically, why do some universities have a defined "graduate school" and associated "graduate faculty," with separate appointment process, and some do not, even when they offer many graduate programs? What is the organizational and administrative thinking behind this?
In US universities, a 'graduate school' is usually an administrative entity that manages all aspects of graduate life. At the University of Utah, for example:
The Graduate School fosters excellence by providing administrative structure and leadership to maintain and enhance graduate education at the University of Utah.
Our programs offer financial assistance, rigorous academic opportunities, and professional development to students, staff, and faculty.
We are guided by the principles of quality, diversity, and integrity as we help students to prepare themselves for successful, relevant careers.
The link posted in the comments suggests that faculty are required to be associated with the grad school in order to function as advisors etc. I think this is also an administrative action, designed to demarcate faculty that can advise students from all the other people who are designated as faculty (for example, adjuncts, research faculty, teaching faculty, and so on).
So there's no academic function associated with the graduate school.
You can think of the graduate school as the university-wide entity that manages the individual chairs of graduate studies at departments.
I am not sure about the US, but in Belgium, Universities are only allowed to give a "real" PhD when they have a doctoral school (equivalent to grad school) attached to them. In some cases this institute can be more silent (background function), and the university is the public face of the PhD, but it HAS TO exist and it has to be recognised. Concerning faculty appointed solely to the doctoral school... well, in my university, the normal faculty where the profs are working has a internal contract with the doctoral school and appoints normal staff to it for a given number of hours.
So in short. It should exist. But it might not be publicly advertised.
At many U.S. schools, faculty must "apply" to be "graduate faculty" in order to do various things related to graduate education: teach graduate courses, sit on thesis committees, serve as thesis advisors, etc. The university's official catalog for each year will have a list of these faculty members. The faculty don't receive anything special (like extra salary) for this.
There is typically a very straightforward "application" consisting of a cover form and a vita that has to be submitted every few years. From the perspective of faculty, this is just a small bureaucratic thing to take care of every few years. It is essentially invisible to students.
Someone in the graduate college reviews each "application" to make sure that the faculty member meets the necessary criteria. But the goal is typically just to document that the graduate faculty are qualified - which already we know they are when they are hired. So the faculty who "apply" are rarely in doubt about the outcome. Faculty who are removed from graduate faculty status typically have stopped being active in research (e.g. on their way to retirement).