dazzajl, What you say is technically correct, from a certain standpoint, but for practical purposes, for the likes of Joe public, including me, to take the same picture from the same spot, and include the same subject/scene at exactly the same size, in percentage terms (not absolute physical size) within the frame being captured, and at the same exposure settings (aperture especially), you will need a different focal length to achieve an equivalent image from each camera. As a result of trying to match everything else in the picture you will find that you get a different DOF from one camera compared to the other.
An example with some made up numbers, but basically sound for the purposes of comparison......
I want to shoot my mate Clive, standing on the other side of the room, about 10' away. He is 6' tall and I want him to fill the frame from top to bottom, with maybe just an inch to spare above and below. On my 40D I find I need a focal length of 50mm to do that. I don't want anything but him to be in focus so I open up as wide as I can to f/1.8.
I now want to shoot an equivalent photograph of Clive with my 1Ds3 full frame camera, from the same position, 10' away, in order to give me the exact same perspective. I find that to keep Clive the same size in the frame I need an 80mm lens, so that's what I use, and I set it to f/1.8 as well.
So I now have an identical photograph from each camera. Clive fills the frame nicely in both and everything was shot with the same aperture and other exposure settings. If I was to print the two shots, which would have Clive looking sharper? Let's look at how the DOF compares....
40D - 50mm, f/1.8, 10' gives a DOF of 0.81' - perfect to keep all of Clive in focus from his belly to his ears

1Ds3 - 80mm, f/1.8, 10' gives a DOF of 0.5' - and Clive's belly is out of focus
So for the same photograph, which is, after all, the things that people actually shoot, you get less DOF from the full frame sensor than the crop sensor, at the same aperture settings.
Your theoretical argument, about cropping out the centre of a print, is not really what photography is about. Nobody shoots to create an absolute size of image on a sensor (say 2mm x 2mm), regardless of the sensor size. People don't go around shooting with the same focal length for the same subject when they swap between camera formats. They change the focal length to suit the format. They shoot to fill the sensor with their subject to the extent that they desire, whatever that sensor size may be. That is the real world of photography. So for practical purposes a smaller sensor does give you a greater DOF, for a given aperture, when you try to create the exact same photograph with each camera.
It doesn't have to be Clive either. Anything will do - a car, a boat, a building, a group of friends, a scenic landscape, a macro shot. Whatever you are shooting, if you want the subject to fit the frame in the same way you will get more DOF at a given aperture with a smaller sensor. What you will find with the larger sensor is that you will have to stop down more in order to achieve an identical photo and that will then mean you have to slow your shutter speed and/or increase your ISO to compensate. That may have implications for camera shake, or subject blur, or noise in the image. Whoever said photography was simple?