Off hand shooting is an acquired skill.
I acquired it as a kid with a BB gun, then a .22 rifle shooting running rabbits and the heads of snakes, and with my 20 ga Parker shooting birds, rabbits, etc. Then shooting skeet and sporting clays.

On occasion I have refused the sticks for African game for a variety of reasons.

I shot my 6x6 elk in Colorado off hand - it was running away at 110 yards. That was with the same Ruger 77 .338 rifle I used on my first plains game hunt.