I think achievements are silly so my hypothetical game would be mocking the concept because if I was a game designer I wouldn't want my game to have any. But they are an expected feature these days so a console should support them. But it should be optional and up to the developer to decide if their game should have them or not. The fact that Sony and MS REQUIRE them is ridiculous. This isn't like making sure the game doesn't crash, this is just dictating content. It's as ridiculous as mandating online multiplayer in every game.
And I don't react well to being forced to do something that should be optional. Thus, if I was a game designer, I would refuse to do achievements in a non-mocking way as a personal protest against Microsoft's ridiculous demands. This would apply for practically anything. If I had to use Kinect, I'd make it so the player has to finger the screen to fulfill the mandatory Kinect requirement. If Nintendo, for some reason, mandated Gamepad touchscreen usage (and they don't thankfully) I would make you draw a mustache on a portrait of an Iwata or Reggie lookalike. If you're going to dictate the content of my game for arbitrary reasons, I'm going to do so in a way that mocks you.