If you mean pixel shift as capturing all three color channels at all pixel locations - it depends, for instance on scene content, but in practice it makes very little difference.
It makes a substantial difference. First there is the benefit of a longer synthetic exposure created by stacking 4 images. It reduces the impact of shot noise in the final resulting merged image.
Red and blue are sampled at twice the frequency and green at 1.4x the frequency. So seams between different colors are captured more distinctly.
As if that were not enough, It all but eliminates color aliasing / moire.
For any static subject where you can use a tripod or camera stand, it is always worth the effort.
As for slanted edge tests, and so on and so forth, and general sophistry in the subject, merely look at the studio scene comparison in DPR with / without it enabled. Don't accept our pontifications on the subject, myself included, DPT gives you abundant evidence as to the practical effect and value of this feature.
As an example, a 400% crop of a micrograph of a flatbed scanner linear CCD sensor. You see the blue and green color filters. Left is standard capture, right is pixel shift ( this is from a Pentax K-1 ) - it is very visibly effective. Its not a gimmick, I use is wherever it is practical.