Yes checkerboarding is a cool technique and it works wonders for the PS4 Pro, that much is certain. Checkerboarding is reconstruction too, just like DLSS but they work quite differently in a lot of areas. Fidelity FX is not comparable to both of these, since that is an upscaling and sharpening technique.
Well if the quality on your 4K TV is good enough with FidelityFX then great, you can use it for a nice performance boost. But the thing is, that flickering/aliasing you see is the difference between a native 4K image (I am aware you are watching a DLSS comparison, see below) and a 1600p image (I assume FidelityFX renders at a resolution like that).
All FidelityFX does is just rendering at a lower resolution, like 1600p, adding a little sharpening filter to make it appear sharper and outputting it as 4K. However, with this technique no new details are added, you play with 1600p resolution, basically. There are a lot of games where you can actually choose your render resolution with a little slider. It's exactly that with the difference of added sharpening (which can be added by turning it on in Nvidia control panel, or AMD's RIS)
With DLSS however, it adds detail to the rendering resolution of just 1440p in Quality mode and it does it so well that it's image quality exceeds native 4K rendering in a lot of areas like temporal stability and detail. DLSS does that because it was trained with 16K feed of games, much higher res than 4K. So there, you truly have a 4K image with the performance of 1440p - around 8-12% DLSS cost. And that kind of feat is truly outstanding and is not achieved by checkerboarding at all.