All of the asymmetries¹ of category theory arise from the fact that we generally talk and think in terms of Set(/Cat/etc.)-like categories and not Set^op-like ones, and in these, everything is a big colimit of simple pieces, but not a big limit.

Vaguer thoughts:

Indeed, in Set/Cat/etc type categories, the categories that we tend to think of as models, generally limits DO obviously exist, but every object has a canonical decomposition as a big colimit of specified primitives (and vice versa, so colimits obviously exist).

On the other hand, there are categories that we think of as algebraic theories, big categories generated by smaller sketches, where every object has some kind of decomposition as a big limit of objects from the smaller interesting stock.

So categories where objects are canonically limits are like categories as (algebraic²) theories, and categories where objects are canonically colimits are like categories of models. (Of course, the former tend to embed contravariantly into the latter via Yoneda embedding, in a manner which takes colimits to limits. But also the former embed covariantly into the latter via any representable functor, which is to say, the former also embed covariantly into the latter via the usual Yoneda embedding, in a manner which takes limits to limits. Hm.)

Footnotes: