Sport coats evolved from riding and shooting jackets. I suspect some mobility has been lost over the years, but neither of those sports requiring raising your arms very high.
Exactly. A lot of what's available to men these days is former athletic/sporting wear turned "athleisure", then sometimes losing much of the original association: button-down collar shirts (there's a reason they're traditionally made of relatively tough Oxford cloth, and the collar's buttoned down so it stays that way during vigorous activity), polos (invented for tennis), sport coats (hunting, riding), "rugby" shirts (obviously).
Then our most popular outerwear shirt, the T-shirt, used to be underwear with a strong military association, and came into common use the same way much of the rest of our modern fashion did: youth culture, which AFAIK is also how we got the blazer (dad's orphaned suit jackets) and two-button and especially two-roll-three suit jacket styles (kids wearing hand-me-down 3 and 4 button suits and orphaned jackets "casually" with fewer buttons done up).
I would hate to shoulder a firearm wearing a modern sport coat. The dimensions of the parts of fabric and the orientation of the seams at the shoulder just don't permit the jacket to fit well while you to bring your hand out in front where it needs to be. I think more than a little mobility was lost (probably in the name of tighter fitment) over the years.
Actually, shooting jackets (for high-power shooting, at least) tend to fit pretty tight. The idea is that they help hold your body more rigid, giving you better stability when shooting at longer ranges (200, 300, and 600 yards). While I agree that sport coats aren't fit the same way, the lack of mobility is pretty common.
Sport coats are descended from the kind of thing well to do Europeans wore wear while shooting clays, riding horses, shooting at foxes while riding on horses, saying nasty things about the slavs, or whatever else well to do Europeans a century ago did for recreation. They aren't descended from shooting specific gear.
Then our most popular outerwear shirt, the T-shirt, used to be underwear with a strong military association, and came into common use the same way much of the rest of our modern fashion did: youth culture, which AFAIK is also how we got the blazer (dad's orphaned suit jackets) and two-button and especially two-roll-three suit jacket styles (kids wearing hand-me-down 3 and 4 button suits and orphaned jackets "casually" with fewer buttons done up).