The first, and perhaps most important, point is that hardly any categories that occur in nature are skeletal. The axiom of choice implies that every category is equivalent to a skeletal one, but such a skeleton is usually artifical and non-canonical. Thus, even if using skeletal categories simplified category theory, it would not mean that the subtleties were artifical, but rather that the naturally occurring subtleties could be removed by an artificial construction (the skeleton).
In fact, however, skeletons don't actually simplify much of anything in category theory. It is true, for instance, that any functor between skeletal categories which is part of an equivalence of categories is actually an isomorphism of categories. However, this isn't really useful because, as mentioned above, most interesting categories are not skeletal. So in practice, one would either still have to deal either with equivalences of categories, or be constantly replacing categories by equivalent skeletal ones, which is even more tedious (and you'd still need the notion of "equivalence" in order to know what it means to replace a category by an "equivalent" skeletal one).
In all the other examples you mention, skeletal categories don't even simplify things that much. In general, not every pseudofunctor between 2-categories is equivalent to a strict functor, and skeletality won't help you here. Even if the hom-categories of your 2-categories are skeletal, there can still be pseudofunctors that aren't equivalent to strict ones, because the data of a pseudofunctor includes coherence isomorphisms that may not be identities. Similarly for cloven and split fibrations. A similar question was raised in the query box here: important data can be encoded in coherence isomorphisms even when they are automorphisms.
The argument in CWM mentioned by Leonid is another good example of the uselessness of skeletons. Here's one final one that's bitten me in the past. You mention that universal objects are unique only up to (unique specified) isomorphism. So one might think that in a skeletal category, universal objects would be unique on the nose. This is actually false, because a universal object is not just an object, but an object together with data exhibiting its universal property, and a single object can have a given universal property in more than one way.
For instance, a product of objects A and B is an object P together with projections P→A and P→B satisfying a universal property. If Q is another object with projections Q→A and Q→B and the same property, then from the universal properties we obtain a unique specified isomorphism P≅Q. Now if the category is skeletal, then we must have P=Q, but that doesn't mean the isomorphism P≅Q is the identity. In fact, if P is a product of A and B with the projections P→A and P→B, then composing these two projections with any automorphism of P produces another product of A and B, which happens to have the same vertex object P but has different projections. So assuming that your category is skeletal doesn't actually make anything any more unique.
You are comparing apples and organges. Model theory should be compared with categorical logic, not category theory. Conversely, category theory should be compared with algebra, not model theory.
Model theory is the study of set-theoretic models of theories expressed in first-order classical logic. As such it is a particular branch of categorical logic, which is the study of models of theories, without insistence on set theory, first order, or classical reasoning.
Best Answer
In topos theory, it is not at all obvious that an elementary topos has finite colimits: there is nothing in the usual definition which would suggest this. And the standard proof, which passes through monadicity theorems and Beck-Chevalley conditions, winds up looking highly technical to most people, and (for most people) doesn't shed much intuitive light on why the result is true.
It's possible to prove this result by other means. There is some information on constructing coproducts in a topos in a hands-on way here, but as one can see it takes a while to set out. One can do this for coequalizers as well, but I don't know where this has been written down (yet).