Funny. I still have an original Nintendo NES, and the only game I ever bought for it (not counting Super Mario Brothers and Duck Hunt, which came with the console) was Tetris. I used to be good at it (I remember scoring high enough to make the Kremlin or whatever take off from the reward screen.) I’ve only played Jenga a handful of times.
I think something like that would work great. I could even attach it to the base’s front lip with hinges so it normally rests on the scraps, and I could just tilt it open so it rests on the sink when I need access.