Your question is hard to answer for several reasons, mostly because you do not define what unique traits you mean. Also, it is hard to assess your level of knowledge in chemistry and physics.

I would argue that mostly it is not due to any packaging effects.

As a biochemist, I consider carbon to be unique mostly because of its nuclear makeup, meaning its position in the periodic table. As first member of main-group 4 it is a relatively small, abundant atom that is capable of forming a variety of different bonds. Meaning sp1-hybridized single bonds, sp2-hybridized double bonds and sp3-hybridized triple bonds. On the other hand, its mediocre electronegativity of 2,5 makes it possible for C to bond with a variety of different other elements more or less stably.

These factors allow for carbon to be perfect as base for organic chemistry under terrestrial conditions - meaning the diverse chemistry of life.