Is there an $\omega$-categorical theory without quantifier elimination? The way I normally prove $\omega$-categoricity (with back-and-forth) immediately gives me QE as a corollary of the test Theorem 13.7 (see picture).
My definition of local isomorphism is:
One example is the theory of an equivalence relation with infinitely many classes of size $1$ and infinitely many classes of size $2$.
Another example is the theory of an infinite graph with a single edge.