For a dedicated 3-gun rifle, It seems that most people recommend either an 18" or 20" barrel with a rifle gas system. I've never shot 3-gun so I'm in no position to question that but I would like to understand the "why" behind the recommendation?

Now if you are shooting irons only then I can understand the longer sight radius benefit but if your running an optic, I can't see how an 18" barrel would have any major advantage over say a 16". I realize the 18" will have a little more velocity but how does that help in 3-gun? The only thing I can think is maybe it's not about the barrel length as much as it is about the gas system?

An 18" barrel provides enough room for a rifle length gas system which recoils softer then a mid-length gas system. Combine a rifle length gas system with a rifle buffer (in a fixed A1 or A2 stock) and one of those competition muzzle brakes and I would imagine it would make for a very soft shooting rifle that would be easy to double and triple tap with? Is that what the recommendation is about or am I missing something?

You got it right. It's all about reducing the sight movement that results from recoil impulse. Couple the rifle length gas with low mass buffer, bolt carrier and an adjustable gas block and you really have a soft shooting rifle.