There is no "correct answer" here except "it depends"....
At a GOLF CLUB, there is not a great need for a green committee provided you have a) a talented superintendent and b) a relationship with a retained GCA.
However, at a family COUNTRY CLUB, they do serve a purpose as much as it can be debated how much that purpose really has impact and benefit.
Primarily, the committee needs to oversee and be mindful of the budget.
Second, as the GM at the club is usually a "hospitality professional" to whom the super generally reports, the committee can assist the GM with golf course issues.
Third, and most common, the committee serves as ambassadors to the membership fielding questions, discussing initiatives and bringing member feedback to meetings. They may be mostly impotent as the board usually does what it wants and uses the green committee as a window dressing component to create the perception of transparency.
"Our committee reflects the diversity here at Whispering Pines CC. We have a junior member, two low handicappers, a club champion, the ladies golf chair, two legacy members, two seniors (male and female) and a board liason."
With the GM, the head pro, the super and the chair, you now have a "bake sale committee" most common at many clubs. Meetings consist of the super giving a course OV, a review of member complaints and issues, a debate about "those trees on #16" and a rant about why there aren't more ball washers on course. Adjourn until the next one....rinse, repeat.
If you have a talented and passionate super with a working relationship with a retained GCA, the committee just needs to get out of the way and empower the process of the professionals. You know you have a club problem when your green committee is way too active and micro-manages the process. That's a sign of weak club leadership and usually indicative that you have a personnel issue.