Center Spin turntable
I would think you could make sure it's centered by having a horizontal object (just a dowel, or something) placed at the center of the item. Turn on the turntable, then look at it through your camera's viewfinder. If the dowel moves back and forth horizontally, then it's not centered on the dowel.

As for fixing in post, it's certainly possible, though maybe a little time consuming. BTW, the "fixed" video in the dewobbler example still has a small amount of wobble that I personally find more irritating than the original large wobble. But maybe it's just me.

To fix it in post, you need to mentally define the center of your object. Then as it moves back and forth, move the center of the object to the center of the frame and set a keyframe. The keyframes will probably create a periodic pattern like a sine wave or something similar. In fact, it seems like that's mainly what they were doing in the dewobbler video - just setting keyframes for moving the center. This can be done in most video editors or motion graphics apps like FCPX, Motion, Premiere, After Effects, etc. Just use the normal transform tool.