How much of the theory needs symmetry, closedness, and completeness

The basic definitions of $\mathcal{V}$-category and $\mathcal{V}$-functor use only a monoidal base (Enriched category over a monoidal base). Symmetry is an extra structure (Symmetric monoidal category), not part of that starting point, and closedness is stronger still (Left-closed, right-closed, and biclosed monoidal categories).

The later items on this page use those stronger hypotheses only when they are actually needed:

• mere monoidality suffices for enriched categories, enriched functors, enriched natural transformations, the underlying ordinary category, and the underlying ordinary category; the strict-2-category theorem additionally uses local smallness so that its hom-categories are honest categories;
• closedness enters when $\mathcal{V}$ is regarded as enriched in itself, when representable enriched functors are formed, and when weights take values in $\mathcal{V}$ itself;
• symmetry is used only where the particular source formula requires it, such as the standard weak-Yoneda setup and the free-enriched-category construction as stated here;
• completeness or cocompleteness of ${\mathcal{V}}_0$ are not needed for the elementary enriched notions, but they do appear in the free-enriched-category and small enriched functor-category constructions.

So the hypothesis ladder is deliberate: later items are stronger because their conclusions are stronger, not because the page flattened everything to one ambient assumption at the start.