One of the many contributions of Morley's work was to introduce a very general model-theoretic notion of dimension. A strongly minimal formula $\phi(x)$ has the property that given any model $M\models T$, we can assign a (possibly infinite) dimension $\kappa$ to $\phi(M)$, and $|\phi(M)| \leq \aleph_0 + \kappa$. Morley proved that models for uncountably categorical theories are completely controlled by the dimensions of their strongly minimal sets.
The proof of Morley's theorem goes like this:
Step 1 (the hard part): If a theory $T$ is categorical in some uncountable cardinal, then there is a strongly minimal formula $\phi(x)$ such that a model $M\models T$ is determined up to isomorphism by the dimension of $\phi(M)$, and $|M| = |\phi(M)|$.
Step 2 (an easy corollary): $T$ is $\kappa$-categorical for all uncountable $\kappa$. If $M,N\models T$ and $|M| = |N| = \kappa$, then $|\phi(M)| = |\phi(N)| = \kappa$, so $\phi(M)$ and $\phi(N)$ both have dimension $\kappa$, and $M\cong N$.
Now the reason that Morley's Theorem seems to add nothing new in each of the classic example cases you have in mind is that in each of these cases, Step 1 is already done, i.e. the strongly minimal set and the dimension notion are already familiar: linear dimension in the case of vector spaces, transcendence degree in the case of algebraically closed fields, cardinality in the case of the theory of infinite sets...
In fact, given any particular uncountably categorical theory, one can prove that it's uncountably categorical without appealing to Morley's theorem by doing Step 1 directly (exhibiting a dimension notion which determines models up to isomorphism) and then giving the argument for Step 2.
The value of Morley's theorem, of course, is that it guarantees that such a dimension notion exists. As such it's very important as a theorem of model theory. It increases our understanding of what the classes of models for first-order theories can look like.
EDIT: I also want point something out about your question. Morley's theorem has the structure "for all theories satisfying this property, the following is true". You complain that you can't find any examples of particular theories for which the conclusion can't be checked without appealing to Morley's theorem. This is a bit like complaining that the Pythagorean Theorem ("for all triangles satisfying the property of being right, the following is true") isn't useful, just because given any particular right triangle, you can do the arithmetic and check that $a^2 + b^2 = c^2$.