What constituents in soft drinks are the cause their strong pungent smell?
Yes, that strong smell is mostly $\ce{CO2}$, possibly with a faint tint of volatile aromatizers from the beverage.

Our sense of smell works in surprisingly non-straightforward ways. Indeed, the smells of the same compound in high and low concentrations are sometimes perceived as quite different, to the point of being totally unrecognizable. A classic example is skatole, which may produce either a pleasant flowery aroma or a fecal stench, depending on the concentration alone. Looks like $\ce{CO2}$ is just a somewhat less extreme example of the same effect.

Air with 5% of $\ce{CO2}$ doesn't smell much of anything (not that it is OK to breathe it for long), which leads most people to conclude that $\ce{CO2}$ is odorless. This is quite wrong. If you try to inhale some pure $\ce{CO2}$ (which I do not recommend or endorse), that's when the smell kicks in. Your breathing just stops instinctively, as if you were trying to inhale water. By that moment, though, some $\ce{CO2}$ reaches your olfactory nerves. I am at a loss of words as to what it feels like. Being slapped across your face with a shovel seems like a moderately adequate approximation. Besides, you feel that familiar refreshing sharp odor of an open Coca-cola bottle.

So no, this is not due to some mysterious additive.