I think there is a flaw in your question, or at least several problematic assumptions about literature, ancient and modern.

Let's take your examples. Firstly, the Bible contain significant portions of poetry (Psalms, large portions of the Prophetic books), but it is not all poetry, and it is not even mostly poetry.

Homer's Iliad is verse, because it emerges in the context of a pre-literate society. Generally, highly oral cultures tend to maintain a high value on poetry and song, because those forms of composition do indeed lend themselves to memorisation. It's much harder to memorise long prose texts, and it's much less interesting to hear long prose texts "performed". Sticking with ancient Greek literature, though, plenty of non-verse material was produced. Herodotus, Thucydides, etc.. The prose genre of history, among others, was "popular". You just have selected a poetic example.

A very similar thing could be said about Ovid's Metamorphoses. Yes, it's poetry. But Romans produced plenty of prose literature - philosophy, for-publication epistles, history - that was popular. At least as much as poetry was.

Dante, I will skip, since it's much later than your other examples of "ancient" literature, and the context of its composition is a different literary world to antiquity.

I think the flaw in your question can be demonstrated simply by reversing the examples: "why was so much prose written in ancient times? The Bible, Livy, Herodotus?"

Verse nowadays seems confined to music, theater and poems.

Which is exactly the same. Your examples are all poetry, that's why they are written in poetic forms. Similarly, plays tend to be written in verse as well (in antiquity). Songs, too, obviously, though our knowledge of ancient melodies is severely limited.