And yet, every DOF calculator I can find changes the results when I change the sensor size. Why? Is it because sensor size is part of the CoC formula?
Yes, sensor size is part of the Depth of Field formula... the output image size, viewing distance, and visual acuity are also part of the formula; but not what the image contains.
The reason sensor size changes the DoField (and the CoC limit) is because a smaller sensor (negative) area has to be enlarged/magnified more in order to create the same size output.
And anything that increases relative size/magnification makes errors (softness) more apparent/visible, which is a reduction in the DoField. That includes shorter subject distances, longer FL's (and TC's), crop factor/cropping, larger image display, and shorter image viewing distance.
Those last two are often left out as variables (and from people's understanding) because they are standardized in the DoField calculation; with the image being viewed from a distance ~ equal to the image diagonal (which keeps its' size relative to the viewer constant)... but if the viewing conditions are not standard then that also changes the applicable CoC limit.
DoField is a perceptual thing and it does not exist as a fixed aspect of an image... Take an image of questionable sharpness and make it small on your monitor, it will look better/sharper with more that is within acceptable focus/sharpness. Then make it larger (magnify it) so it looks kind of bad and things that are not sharp/in focus are quite apparent. Now get up and walk away until it looks good again... that is DoField in action.
Changing sensor size does not change the Depth of Focus; which is the relative focus/point sizes as recorded at the image plane (the focus tolerance).
Edit to add: that video is confusing Depth of Focus/Focus tolerance as recorded at the image plane with Depth of Field in the final image/viewing condition. They are intricately related, but they are not the same thing.
Or to put it another way, because he cropped the FF sensor much harder (much greater magnification) in order to get the same remaining image as the 4/3 image, he created the same DoField in the end results... in this case the original format CoC is not applicable because that relates to enlarging the whole image area; they would instead have the same CoC limit because they are the same portion of the image circle projected by the lens. Cropping in post has the same effect as cropping at the time the image is taken (using a crop mode/crop sensor).
As I said at the beginning, the DoField formula does not care what the final image contains (output field of view)... he did the opposite. But, TBF; quite often people are really talking about DoFocus as an inherent aspect of an image (which it is) when they talk about/say DoField... they just don't know it.