I'm a member at Beverly and Olympia in Chicago where numerous trees have been removed. Some were removed for turf reasons. Some were removed because of overplanting or because they negatively affected the playability of a hole. Some were removed because they were a terrible species (i.e. silver maples which deposit all manner of trash on a course and which have very shallow root systems that plague turf and mowers). In every, single instance, IMO, the holes look much better after tree removal. I was totally shocked when we removed twenty or so mature oaks around the fourth green at Olympia because grass wouldn't grow on the green. We were very concerned that the hole would look worse, but were amazed to see that the green complex really stood out without the trees.
Trees are emotional hot-button issues for golf club members. Nobody, after all, ever wrote a poem about a stump. But the guiding principal for anybody in this situation is to listen to the experts. Listen to the architecture experts, listen to your grounds superintendent and listen to arborists. You'll take a lot of heat for removing a lot of trees, because the tree huggers are an understandably vocal lot, but you'll always improve your golf course.
Finally, nobody is seriously advocating removing every tree on old golf courses that were built on largely treeless sites. The more sensible tree removal advocates are looking to remove trees that hurt important turf (tees and greens), trees that negatively affect play and trees that never should have been planted on a golf course (silver maple, for one).\
We're talking about golf courses, after all, not arboretums.