From an optical point of view, light does not bend differently just because you put a differently-sized rectangle somewhere in its path. Or to put it another way, if you cut the edges off your sensor, that won’t alter the image on the remaining area of the sensor.
Only in these sense that you generally use a smaller sensor because you want your camera to be small.
If you take a full frame SLR and attach a 100mm f1.8 lens to it, you’ll get a shallow depth of field. Now crop that image down to an area of the sensor corresponding to the size of a phone sensor, and the cropped image will have the exact same depth of field.
> Now, here's the kicker: the bigger the focusing lens is, the larger the cone of light rays is, meaning the the out of focus parts of the image will be more out of focus
From the page [0] it takes the depth of focus image from:
> [Depth of focus] differs from depth of field because it describes the distance over which light is focused at the camera's sensor, as opposed to the subject
[0] https://www.cambridgeincolour.com/tutorials/depth-of-field.h...
Depth of focus isn’t really relevant to the rendering of an image (except insofar as you want your camera to be built to sufficient tolerances that a sharp image can be obtained when desired).
I assumed you were using “depth of focus” to mean “depth of field”. If you really meant “depth of focus”, then I would say you are mistaken in thinking that the author’s goal is to obtain a narrow depth of focus.