I concur with much already commented on by fellow superintendents on this site. However, I will say this:
The courses with higher expectations, and commitment to growth and improvement, which often offer the higher salaries for superintedents, should want and get a more "executive" type superintendent.
I've worked for a couple of these higher-end private facilities, and was never uncomfortable with the fact that my boss always dressed up in nice slacks and a tie. The mentors I learned under showed me their expertise in this field by expert hiring and finding the right people to do the "dirty jobs" for them, then continue to teach and train those like me working up the ranks. The boss never needed to get dirty, but I knew he knew his stuff because there's no way he could manage the persons who did train me without being an expert in being able to evaluate the work they were doing.
That being said, I'm also personally experiencing this dilemma now at my own course. We're not higher end, but we have had healthy budgets in years past and have always invested in capital and improvements. With the current cutbacks, my assistant has become my mechanic, and I've taken over most all normal assistant responsibilities. Having to get out and get dirty more often (though I don't mind it at all) has given me much less time to focus on improvement projects and future planning, and required that I just be satisfied with maintaining the higher day to day course detail expectations and less on needed and sometimes necessary improvements to the course. Should times get better, I would hope to be able to increase my staffing levels, put people I've now trained into trainer positions, and give myself more time to focus on planning and overseeing course improvement projects. But for now, I'll keep fixing irrigation leaks and spraying and mowing greens as my more primary job positions.
Clubs that can afford to hire a superintendent with the skills to lead by example, with enough budget to hire quality crew leaders for key positions and free their time up to looking ahead and striving to constantly make improvements to the course as opposed to a course that can only afford (or even just wants) to hire "a superintendent that likes to get dirty" will definetely reap more benefits.
We all like to get dirty, it's the nature of our job, but it's much more fun to roll around in the the topsoil of a new construction or rennovation project you've planned than the mud hole of an irrigation break you knew was going happen but never had time to correct because you were too busy having to mow fairways.