There is no DOF advantage of smaller formats. Unfortunately, this myth seems to persist because at first glance it is true. But it isn't. No format has ANY DOF advantage over any other format once you factor in diffraction.
For a given FOV, lenses with the same physical aperture (in mm, not f-stops) will have the same DOF and diffraction characteristics on any format, large and small.
That's an interesting point, but it does need to be emphasized that there is no DOF advantage of smaller formats
provided that everything else that is relevant remains the same, including the FOV in the image, the pixel pitch, the pixel quality of the sensor, and the lens quality and characteristics.
An obvious example would be the comparison between the 20 mp Nikon D500 (a cropped format) and the 45 mp D850 (full frame format). Both cameras appear to have the same quality of pixels (according to DXOMark), and the same density of pixels. The Nikon full frame sensor is 2.25x the area of their cropped DX format. 2.25 x 20 = 45.
Suppose we use both cameras with the same 50 mm lens, shooting the same scene at the same f/stop, using the same ISO and the same shutter speed. The image from the D850 will have a wider FOV, so in order to compare apples with apples it needs to be cropped to the same size as the D500 image. The cropped D850 image will then be identical to the D500 image, excluding the normal margins of error of course.
It will have the same DOF, same degree of sharpness, same amount of noise, same degree of overexposure or underexposure, and so on.
The smaller format in this example would appear to have no advantage at all, in terms of inherent image quality, although it clearly has an advantage in respect of weight, and possibly other software features which I haven't investigated.
However, the larger format of the D850, in this example, has the very obvious advantage of turning a standard 50mm lens into a 50-75mm zoom, from the perspective of the smaller DX format, and a zoom which is likely much better quality than the equivalent 33-50 mm zoom which would give the D500 the same range of FOVs as the D850 with 50 mm lens.