The "look" is a great thing to discover in each and every property as it relates to a variety of regional settings. Diverse land and country side have their own unique "looks" to offer. I don't think that it is necessarily effective to try to create the sand dunes or links look in a flat cornfield in Indiana with horizons that are totally out of whack with the 200 or so acres where you have dozed up a bunch of hummocks, dunes and bunkers. The interior of the course has to somewhat blend with its greater surrounds.
Mr. Huxford, if your club wants to cut down every tree on the course, yet the surrounds are to some extent wooded, sparsely or heavily, I think that it would not be effective to remove just the trees on the golf property. You have said that the few trees that are there don't encroach on the lines of play. Then, I believe that if natural habitat for birds, and ground/tree dwellers is provided, they should be left as is and trimmed when needed.
I am thinking of the "look" of some of the Australian sand belt - heath land courses we see in the winter months when there are tour events there. They look fine with trees and shrub vegitation adequately cleared away from lines of play.
Here in Wisconsin, we have a links-style course in a broad clearing of mostly wooded and lakes land. There was an extensive program in 50s and 60s to add trees to the once links-like concept of Langford from 1930. In a style much like Raynor, Langford manufactured a links-like emmulation. It plays and fits very well because he did not try to actually create a true links environment, only a suggestion of similar characteristics. There were areas of trees that did not encroach play that were left to blend with the surrounding countryside in Langfords original. Then with the tree planting programs, the quality of the intended style of play was seriously deminished. Now, with wise guidance from Architect Ron Forse, the Lawsonia golf course operators have cleared the planted trees out and returned it to a very nice setting and within the context of links-like playability.
That is the proper way to go in my opinion.