Personally, I think making the path and beyond OOB could actually improve the hole as well as being a sensible H&S measure. In essence, the path would be the mirror image of the stone wall on #14, and serve exactly the same function. While not on the perimeter of the property, the road marks a logical boundary of the playing area and so would not fall into the dreaded category of "internal OOB".
On reviewing the best photo I could find of the tee shot, I am beginning to concur with your idea of trimming back the trees, at least to reveal a view of the clubhouse and more of the road.
18b by
duncan cheslett, on Flickr
The photo is not a perfect representation, as it is taken from well in front of the daily tees. The medal tee is some 80 yards further back, in an Augustaesque tunnel of trees.
And yes, the clubhouse does get hit on occasion. Warning signs are prominent on the terrace that members and guests sit there at their own risk.
Only last month a ball went straight through the boardroom window on the second floor. It was hit by a visitor - a left handed slicer. They seem to cause a disproportionate number of problems!