I don't use a T/S lens, but I do shoot with bellows, that will do the same thing.
The tilt part is, as you say, primarily for DOF control. Or to put it more accurately, to control the plane of focus. By tilting the lens it is possible to move the plane of perfect focus (if I may term it that) away from being perfectly parallel to the film plane. With landscapes possessing both near and far elements, moving the plane of focus and stopping down partway gives sharper images than keeping the lens perpendicular and stopping down a lot.
That being said, I've never really felt the need to tilt a lens as short as 24mm, as f/11 focused at the hyperfocal distance (5 ft) gives 30 lp/mm from 3 feet to infinity. For longer lenses the tilt capability becomes more important.
Shift, moving the lens sideways or up/down, is needed for perspective control. If you are shooting a cliff face, for example, with a rigid lens you need to tilt the camera up to see all of it, which places the film plane at an angle to the cliff. When you do this, the image on the film will keystone, making the top of the cliff appear to receed from the base (to lean back, as it were). So landscape photographers always try to keep the film plane vertical (or parallel to the object), and if we need the camera to "look up", we instead shift the lens up so the top of the cliff is visible, but leave the film plane still vertical. Same thing applies to shooting something from the side, at an angle. If we point the film directly at the object, then shift the lens sideways, we can see the object as it is, and the distant part will not look smaller than the near part.
I should say that most of us, when we recognise the need for tilts and shifts, move to a 4x5 bellows camera. They are great for this sort of thing. Nikon used to sell a bellows for 35mm cameras, but it aded length to the lens and was really only good for macro shots (where, because of the short object-to-lens distances and consequent miniscule DOF, DOF control is very important).
Added in Edit: I forgot to add that for wide-angle lenses like the 24mm lens having shift is very important, as a tilted film plane using a wide angle shows much more convergence than a normal or long lens.
Also, to be obvious, lens tilt controls the angle of the plane of perfect focus, and lens aperture controls the depth of focus on either side of that plane (in perpendicuar direction), however it is oriented. By tilting the lens down enough you can get perfect focus of the desert floor with the lens wide-open, and stopping down the lens would bring tall plants into sharper focus. Here is an example: http://science.uvsc.edu/chemistry/wilson/p...p?Photo_ID=6632