As a junior overseas member of a nearby club for seven seasons, the last four a bit frustrating, I remain deferential to the local members who feel the impact of operating decisions much more directly. These good folks enjoy the full benefits over the entire year at essentially similar costs, so, though somewhat grudgingly, I've come to accept decisions which accrue to their benefit even at the expense of my own.
Brora is special. I've played the course more than a dozen times over the years and always look forward to returning. The animals have never bothered me in the least, though stepping over the electric wire strands is becoming more annoying and difficult. I particularly like the juxtaposition to its busier, more regulated, highly-acclaimed neighbor. Brora as well as Golspie, without sacrificing the core golf experience, feel more relaxed, with no feint of pretension, and appreciative of my visits.
Undoubtedly, a small number will be put-off if the grazing animals are moved from the property. I don't see the cows or the sheep to be integral to the golfing experience. I know that it is dangerous to generalize the experience gained from 60+ years of maintaining my residential landscape to 100+ acres of golfing turf, but I've seen the negative effects rabbits can have eating and releasing their waste in small areas. There is a reason farmers rotate their livestock and considerable anecdotal evidence of large herds of wildlife devastating areas before moving on to do the same elsewhere.
I suspect that removing the animals will not have a negative financial impact on the club and could very well contribute to improved playing conditions. Southerndown in Wales is the only course I can think of whose livestock added to the playing experience.