I actually feel quite lucky to be a member at 2 courses that (currently) have sheep on the course, but both in totally different ways. Brora has already been well discussed, but let me mention the sheep at Hollinwell as a comparison...
Last summer, when most courses in the UK were struggling with heavy rough due to the amount of rain, I hit a terrible tee shot on the 13th at Hollinwell (long downhill par 3) well out to the right. This went over an internal fence, to an area that was in play. I went over the sty, and my ball was easily found and sitting reasonably well, allowing an up and down for par. Anywhere else on the course that wasnt fenced off and I'd have probably struggled to find it, let alone make a par!
The fences are there as we have a herd of sheep kept at the course by the Notts Wildlife Trust, but this is helping with our heathland management and not necessarily for golfing or agronomy reasons. There are quite a few of these areas around the course, mostly around the edges or out of the way, but the one on 13 is the main one in play, and its clear the sheep help keep the rough in a condition in there that is sensible and not brutal.
We also recently temporarily fenced off an area for sheep around our waste area / scrape / bunker on the par 3 5th. This was because the decision was made to make the waste area more natural, and the sheep did an excellent job of breaking up the more manufactured edges (technically damaging bunker edges, but that was the natural effect desired in this case) as well as thinning out some of the lusher grasses around the edges.
After the work on the 5th I did mention to the course manager we should do this elsewhere on the course, but the brief discussion was along the lines of the members are okay with the sheep out of the way but wouldnt go for it closer to the playing areas, but in theory he would be happy to have more sheep as long as they were just in the rough. And I think thats the crux of it, the sheep at Hollinwell are small numbers and out of the way, so the issue of their droppings isnt a problem.
I mentioned on the other thread that as a Country Member at Brora, I didnt get a vote. However, I have played it enough to understand that while there are lots of wonderful reasons for having the sheep, be it agronomy (some mentioned above as at Hollinwell) or spirit of place, it doesnt matter how much time the greenstaff take shovelling sh*t to make the fairways clear, your ball or shoes are going to come into contact with it. And while as a visitor that may only happen occasionally and be a novelty, I can understand why a member who plays regularly, would vote to not have to wipe sh*t off their shoes or golf balls.
Cheers, Jamesps with so many people apparently travelling all the way to Brora just for the sheep, perhaps if we ever need to increase visitor income at Hollinwell we just need to change the marketing strategy to focus more on pictures of the Herdwicks!