AppleTV's Netflix browse menu is no longer accessible
By 'browse interface' do you mean the view where you have a grid of cover art thumbnails? 

After you log in to Netflix, you can go to "Suggestions for you" from the menu, and you'd be presented with the grid, customised to your favourite genres and recommendations based on what you've watched and liked. The TV Genres and Movie Genres are similar but each row is a different genre.