Would it have been possible for Native Americans domesticate mountain goats and big horn sheep, or is there something about them that makes them harder to domesticate than other goat/sheep species?

Others have mentioned the difficulties of catching and domesticating mountain goats and Bighorns.

I’m going to look at what the benefits of Bighorn sheep might have been to the Native Americans. Sorry, I don’t know much about goats.

Factors in sheep domestication:

Milk
Meat
Fleece
Skins
Docility

(1) I don’t know if Bighorns are particularly good milkers, but, if domesticated, the better milkers could be selected for. However, at least 74% of Native Americans lack the gene mutation for adult lactose tolerance. That figure could well have been higher in the past, so they would not necessarily have wanted to domesticate Bighorns for milk.

(2) Bighorns are big buggers, and can weigh up to 300 pounds. This would be a good reason for their domestication, but absent other reasons, why not simply hunt wild Bighorns?

(3) Bighorns are hair sheep. They have minimal undercoat (fleece).

(4) Another good reason for domestication, but insufficient on its own: hides would be a byproduct of hunting for meat.

(5) Bighorns are BIG. 2–300 pounds, around 1 metre (40 ins) tall at the shoulder, 1.6 – 1.85 m (5–6ft) long nose to tail. Those horns alone can weigh 14kg (30 pounds). This alone wouldn’t render them undomesticable – cattle are much larger. However, combined with their social structure, their size alone is risky. Unlike domesticated sheep, they live communally – males and females together, rather than many females and one ram. The latter social structure works for domesticability: if you can control the ram, you can control the flock – or simply eliminate the ram, and the ewes will follow you as the new leader. With Bighorns, you’d need to eliminate all the rams – even keeping one for breeding doesn’t keep control of the flock – and gain control of the oldest ewe (female Bighorns have an age-related hierarchy). If you did somehow manage to keep a Bighorn flock whole, you would have to deal with big, dangerous rams attacking each other (and, presumably, you) in rut to establish a pecking order for access to fertile ewes.

Herding Bighorns is far too much effort for access to their meat and skins – which can be obtained much more economically through hunting.

Quora linky.