As Jesse Schell says in the book "The Art of Game Design: A Book of Lenses" (DEFINITELY READ THIS BOOK), games are an experience. Mechanics are certainly a very important part of a game but they are just that - a part. They can be a bigger or smaller part depending on the overall experience you want to deliver. Mechanics alone rarely stand though. Even if you don't care much about the story or other elements it all still needs to be there to some degree (or absent for an intentional reason) for a complete experience.
Let's use your example, "Threes", to demonstrate. "Threes" is a game where you have cards on a grid, you slide them around, and you try to match together powers of three. There is no written story included with the game. It's mostly pure gameplay. However the experience and the themes that are set in "Threes" are what make the game what it is. The music, the minimalist style, the emotions expressed by the little cards and the sounds that come with them do create a story that is as equally minimalist as its visuals. If you took that away and just had cards with numbers that slide around it wouldn't be "Threes". The little things like two cards greeting each other make for a small story of, for example, two people trying to get to each other in a crowded room, that just wouldn't exist otherwise. That's a story the player interprets but that doesn't make it any less important. That's part of the experience of the game. Likewise, Bioshock or Ocarina of Time would just not be the same or as fun without their stories and themes - even if you care little about them.
Mechanics are focused on so tightly because it's one of the only elements in video (or tabletop) games that make them stand out from other mediums. However if you pay attention to games that are often well regarded you'll notice there's always more to them then pure mechanics. They are always an experience that's usually supported by strong mechanics.
I could also get into how the story itself can become a mechanic (Shadows of Mordor) but I think if my comment gets any longer it's just going to be too much to try and feature. But Shadows of Mordor...yeah there ya go.