You are correct.

For a moment and think about it one piece at a time.

Assume you have a camera where you can change anything you please. For large format cameras this is actually the norm.

If you have a given subject say a person and you want them to be a certain size in the frame, your camera is on a tripod and all you change is the format what happens?

The subject doesn't fit the same way anymore.

To fix that you can;

Move the camera. Getting closer while using the same lens and aperture reduces our DOF and vice versa.

Change the lens. Here longer lenses have shorter DOF at any given aperture and vice versa.

Once you fix the issue above then you adjust aperture.