I used the Taxonomy View Depth module to create a taxonomy vocabulary like this.
What I did was enabling the default taxonomy view, and changing the depth in Content: Has taxonomy term ID (with depth) to 1 instead of 0 (also tried 2,3, etc.)
The children items will never show on the page. It does remove the There is currently no content classified with this term. from the page.
What I want? With Taxonomy menu I want to be able to click Cats and show the second level children.