Can everything be made cardboard-compatible?

No. There's a lot more to making a VR game than just displaying existing 3D content in a different mode, and not all existing content is even 3D. Your Canabalt example is an obvious case that won't work for the simple reason it isn't a 3D game. There's a lot of extra work needed to compute the extra view for a second eye, and that's a source-code change, not something that can be done automatically.

Even the 3D first-person games that would lend themselves to VR conversion aren't that simple. PC VR fans have tried patching graphics drivers to enable this with mixed results. The problem is, many games use some parts that aren't a simple 3D view: a 2D HUD, or drawing the first-person weapon in a different pass with depth disabled. These features all cause problems, and require game-specific fixes. On top of that, even when you make your target game "just work" in VR, it'll probably make you throw up. Interaction and motion in VR need extra thought to make immersion work without causing motion sickness.