First to get it out of the way, nothing can 100% accurately reconstruct missing pixel data. However, rendering more pixels has a roughly linear cost while giving ever diminishing returns to what the eye can see.
1080p is already really close to optimal resolution at normal view distances. Native 4k detail improvement is quite marginal without super sizing your TV-to-view-distance ratio. Rendering 2k and doing a high quality upscale is going to give you a majority of visual improvement that native 4k could. It's just a law of diminishing returns. Rather than using 4X the power to render 4K you can use the power for other enhancements that are more easy to see.
The checkerboard solution is a good way to use half the pixel coverage with uniform scaling in both X & Y axes while minimizing softness. Also the checkerboard pattern may be alternated each frame to get a higher temporal resolution (similar to interlacing). It's already been done with great effect. Nice one Sony if they have incorporated 'checker' rendering in hardware.
