"Control" means something different every time the word is used in the coaching context. Possible elements include any combination of these:
(1) Recruiting -- freedom (or lack of freedom) to waive academic minimums, "whole person" considerations (e.g., disciplinary issues), decide unilaterally versus in a negotiation with the dean of admissions. How autonomous is the coach in his ability to invite athletes to become students at the university?
(2) Scholarships -- freedom (or lack thereof) to decide on scholarship awards; often goes hand in hand with recruiting, but is a separate issue.
(3) Scheduling -- the team calendar can be multi-faceted: not just games, but practices, spring and fall camps, community activities, team building trips, etc. The academic calendar has some big hard pillars built into it, like move-in and move-out windows, term end exam periods, graduation dates, proms, campus activities, and so on. Melding those two calendars together can be a real challenge. So how much autonomy does the football coach have in deciding how his schedule will interact with the larger one?
(4) Hiring and Firing -- most coaches want to have at least a deciding role in choosing their assistant coaches, S&C staff, medical and nutrition support staff, and so on. Some want full autonomy in these decisions, in spite of the fact that many schools have to abide by complex federal and state legal requirements in the hiring and staff management processes. Some coaches even want say over hiring and firing in parts of the AD staff, those parts most directly supporting uniquely the football program. It's a complex business, and one that begs for control issues.
(5) Finances and Resource Management -- a lot of coaches want to have nothing to do with the business side of the football program, are more than happy to let the Athletic Director's staff manage it. But some want to control parts of it, or even much of it, from the immediate (purchasing team equipment, setting and adjusting staff salaries) to the long term and strategic (improvements to stadium and team facilities, management of real estate, construction of new facilities, student athlete dorm considerations, the list is almost endless if you dive into the weeds).
(6) Academics and Academic Support -- there's this whole question of the extent to which student athletes can receive support from staff in their academics. Tutors, and aides, and secretarial staff, and so on. Some coaches like to get deeply involved in these questions, while many academic deans prefer the athletic leadership stay at arm's distance.
(7) Discipline -- players (and sometimes, assistant coaches) are going to get in trouble from time to time. How autonomous is the head coach in deciding punishment? Does he have to enforce and follow standards established for the university at large?
(8) Communications -- who deals with the media? And how much freedom do they have to interact however they wish? Will the coach's contract require him a certain number of media interactions throughout the year? And does the coach have to clear his talking points and messages with the university leadership? On and on this one can go.
(9) Outreach and Representational Duties -- the head football (and/or basketball) coach at many universities is the rock star of the faculty, often the only name known by the majority of folks who live in the state. So how much is the coach required to represent the university at social, community, fund-raising, alumni, and other events? This can be a big part of a head coach's job, measured by amount of time committed.
These are some of the bigger lumps in the oatmeal, but there's a lot more as well. The question of "control" is linked closely to the question of "responsibility," and while most coaches want as much as they can get of the first, they often do not want to be saddled with too much of the latter outside the football stadium. So it's a balancing act, and every university handles it all a bit differently.
p.s. At most universities, most of the time, boosters have little to no control over anything. They have voices, and often use them, and try to bring pressure, but control = hands on the steering wheels...and that, they pretty much never have. So all the above isn't about boosters, by and large.
