Hello,
here is my opinion on this topic, but I believe this requires a complete separation between keyboard or touch screen users.
Keyboard
If you use a keyboard, in my view, this kind of thing is completely useless and just complicates the matter. Of course, you may not be aware, so just in case, the list of games supports multi-letter navigation. So, keeping this in mind, it's much faster, if you want to play Farkle, to choose create a new table and type a few letters, in this case FA is enough since there is no other game starting with FA at the moment, and you're on Farkle, that's a lot better compared to clicking on Create a new table, then finding some dice category, entering that, and then finally finding Farkle.
With the Farkle example it isn't even so bad, but it's not always so clear and obvious which category a game should be put into. You may find yourself entering a wrong category because, say, you expected Duck racing under board games while I consider board games only the ones played on a grid (this is a matter of English ambiguity).
Then there is one more problem, some categories don't even have enough games to warrant the separation. For instance, where would you put Quiz party? Logic would dictate it would go under some separate trivia category, but there aren't any other trivia games, so making a category for it alone seems like a stretch. At most, you could really go far and say put it together with the Little exam, but they aren't similar enough.
Finally, it's worth noting that even in the current list of games, games are often sorted so that similar games are next to each other (you'll notice in the list we have Farkle, Yahtzee, Zanzibar and Shut the boxes all games focused on dice next to each other, trick-taking games are grouped together next to each other as well, and so on).
Touch screen users
Here, it gets a little more interesting, and I can see this being useful. But again, I personally don't see it as useful under the suggested form, enter one menu, try to find your game, realize it is not where you expected it, exit that menu, enter some other menu. Even under the ideal form where you know the category of each game and everything is completely obvious, I really think those kinds of nested menus just make the table creation unnecessarily slower, or at least they don't really significantly speed it up in any way.
Where I would really see this as a great design is if these categories were headings on the Web client itself. You could then jump between headings to quickly navigate if you want to pass a particular category, you could easily swipe back and forth between them, and so on. Unfortunately I don't think the current layout of the Playroom really allows doing that sort of thing easily, but it might be worth thinking about it for the future.
By the way, there was a bit of discussion related to this in this topic.