Getting a goat tag in AK or British Columbia is pretty easy but a pricey hunt. On average you have a better chance getting drawn for elk in KY every year than getting a sheep tag in MT or CO. There is a hunting district in the Missouri Breaks area of Montana that gets 9000 applications for 6 tags.
Regarding a mountain rifle, weight is the most important factor. You want a very light rifle that is carried a lot and shot a little. All three of those cartridges are more than capable of taking a sheep of goat at 400. You just want to make sure you can anchor the animal. I saw a goat that went down a 1000' scree slope. It took days and rappelling gear to get him out.
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