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I think yall are basically both correct, but simply viewing the equation from two different angles.

Example. Let say I have a 12MP full frame sensor camera and a 12 MP and am shooting with my 100mm macro. OK, if I am at the min focal length that lens will fill the sensor with a 1:1 macro but the FF sensor is bigger so that image will have more of the scene in it as opposed to the crop sensor. The DoF at the min focal length is going to be exactly the same. But, if I want the same image with the crop sensor I will have to move back beyond the min focal length and by moving back I will increase my DoF basically giving the effect of an increased DoF.

It is really the same thing as thinking you get more reach with a crop sensor. Technically, you don’t, it is simply a cropped image but the end effect is like adding a factor of 1.6 or 1.5 depending if you are a C or an N (or 1.3 for some 1Cs)