Can skeleta simplify category theory?

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.

I think that one of the basic ideas of category theory, and of much of current trends in various areas, is that one should not fight the fact that there are choices by actually making choices, but instead that one should deal with them all at the same time, for the presence of choices is itself an important piece of information.

In MacLane's "Categories for the Working Mathematician", in the end of section VII.1 there is an argument due to Isbell proving that one cannot strictify a monoidal category by replacing it with its skeleton.

Generally, I would say that all one can achieve by replacing a category with its skeleton is replacing natural isomorphisms with much less natural automorphisms.