Sheep are naturally a dryland animal. In the past in the UK they were an animal the dry eastern and southern parts of the UK.
It was only the Highland clearances where landowners booted their tenants off to North America to make way for sheep that they came to the wet north western parts of the UK. And they never did particularly well commercially and once we had opened up Australia and Argentina sheep production for wool went elsewhere. The lairds found it more profitable to lease / sell their lands for deer and grouse to the new wealthy of the late Victorian period.
Sheep only really survive these days in the highlands thanks to subsidies.
And now thanks to changes in Government policy, deer are now viewed as a pest and we as tax payers are spending millions on shooting them.
I can’t help feeling that reintroducing top predators - which are struggling elsewhere, that can then manage deer, rabbit, hare, beaver and other smaller predators is probably a much better use of tax payers money.
But you will need to manage top predators and occasionally need to shoot them when they get too close to urban / intensively farmed areas.
And once you have large predators they will not tolerate smaller predators such as foxes, badgers etc and hence you may actually end up with lower overall pressure from predators.
As to Capercaillie and Black Grouse populations, the biggest killer of these are deer fences, whether from flying into them, or being trapped against them by foxes etc. They seem to do pretty well in Scandanavia, Russia, Alpine areas where they are living alongside Lynx, Wolves and Bears.