From the perspective of UK private members clubs, you tend to have committee's galore and 'too many fingers in the pie'. The folk on such committee's, who are normally retired and although well intentioned, they generally tend not to have a clue about golf or how to run a golf club, not a damn clue. Donkeys leading lions. The joke "Why do golf clubs always have showers in the lockers rooms?.......Because a Committee cannot run a bath" is not entirely in jest. The strength of the Director of Golf role as I see it is, if the club is set-up correctly, ie with an appropriate set of rules and constitution etc, that a DoG focuses authority and responsibility in one place and reduces the number of fingers in the pie and the impact of well meaning but utterly useless, frequently harmful, committee's.
In addition, traditionally golf club Secretaries tend to be folk from non-golf backgrounds who have retired early and take the role on as a 'fill-in job' until full retirement. They also quite likely have an hidden personal incentive/agenda of free golf for life thereafter as an Honoury club member. A Director of Golf, an ex-pro with business nous and organisational skills on top of his/her professional knowledge of the game and shop skills etc, can reduce the impact of too many fingers in the pie and reduce the impact of too many committee's. Plus, if the DoG oversees the Head Greenkeeper, then someone who knows how to play golf and how a course ought to be set-up will have authority to ensure the course is prepared properly and independently, especially if the head of the maintenance crew, as does happen, is a non-golfer. It also avoids the Chairman of Greens fiddling with the course to suit the golfing ability or tastes of himself and his mates, and this does happen.
All the best.