In which I question the NEX APS-C Shallower DOF argument

The diagrams and Blue_skies are correct.

Essentially DOF (depth of field) depends on the three factors Focal length, Distance to Subject and Aperture.

However sensor size is implicitly a very important factor as it only makes sense to compare DOF at same field of view (FOV) as Blue_skies were saying.

Example: If I pop on a 35mm/f1.8 legacy lens on a NEX and the same one on the m43 then we will roughly get the same DOF at the same aperture. However, the FOV will be very different as it will be 52mm and 70mm respectively.

So to actually get the similar FOV you would use a say 25mm on the m43 and a 35mm on the NEX, and suddenly the potential of isolating the subject gives a big advantage to the APS-C sized sensor.

Moreover, I believe a bigger physical aperture size (haven't studied this) will more effectively throw away the background if needed. So...

...if we are using a 25mm/1.4 on the m43 the aperture size is --> 17.9mm

...a 35mm/1.4 on a APS-C body will give an aperture size of --> 25.0mm

Again the APS-C sensor has the advantage.

In summary, a, APS-C sensor camera is more effective to isolate the subject (or blur out the background) than a m43 sensored camera.