Changing format size does affect DOF, and it affects it in the opposite direction that you think it would.
If you hold all things constant (focal length, aperture, focus distance, print size) and only change the size of the format, then the larger film format will have more depth of field than the smaller format. This is because the larger film format requires less enlargement to reach the desired print size, and therefore uses a less stringent COC.
It is only when you change focal lengths to match the FOV between formats that the larger format has less depth of field. The effect of lens focal length overpowers the effect of format size when calculating DOF.
A simple way to state it is:
Larger formats require the use of longer focal length lenses to accomplish any given photographic purpose, and that use of longer focal length lenses leads to functionally reduced depth of field.