Clearly DMOZ hasn't been getting a lot of real user traffic, which I think is because newer users (that's most users) have only ever used search, never been introduced to the concept of a directory.
So, the focus should be on who it's being built for, and how they would find and use it, rather than just replicating something from a different era.
For example, it could be a blog where guest bloggers (or blogger teams) each submit a carefully edited post describing the top sites (and perhaps apps) in a niche, and what's good about them. And then keep the posts up to date.
Quality could be maintained by a moderating team that only accepts the best posts, and ensures that they are maintained (otherwise deleted).
The resulting post when found in a search engine would be more readable, and probably more useful, than a directory listing page.