But I don't understand why F2.8 is such a "magical" turning point.

There's not any such magical turning point in theory. But in practice many camera systems with different lens mount designs and even vastly different registration distances (or flange focal distances) tend to have near the same throat diameter. The throat diameter is the width of the hole in the middle of the ring to which an interchangeable lens attaches.

The reason throat diameter is a factor is because even if lenses only need entrance pupils (the aperture as seen through the front of the lens) smaller than that allowed by a lens with diameter just large enough to cover the hole in the front of the camera the lens will still be slightly larger in diameter than the throat diameter. The need for the diameter of the lens barrel, at least at the rear of the lens, to be larger than the throat diameter should be obvious.

For the typical throat diameters of most common SLR/DSLR and even some mirrorless systems, that minimum lens barrel diameter roughly corresponds to the diameter needed to make a normal zoom lens (a lens with focal length encompassing between about 0.65X to 1.5X the diagonal of the film format or digital sensor) with a maximum aperture somewhere around f/3.5 or f/4 at the wide end and about f/5.6 on the long end.

With wider angle lenses the angle of view is just as, if not more, important than the maximum aperture in determining the minimum size of the front element of a lens. But until one gets into ultra-wide angle or telephoto territory, a lot of typical lenses still need front elements about the same size as that of a lens barrel large enough to cover the throat opening of the cameras for which they're made. To get constant aperture f/2.8 zoom lenses in the same focal length range, the front elements need to be larger than the "minimum" lens barrel size, as determined by the throat diameter, of most SLR/DSLR/mirrorless camera systems.

That's one reason kit lenses tend to be 18-55 f/3.5-5.6 for Nikon, Canon, Sony, Pentax, etc. crop bodies. For an 18mm APS-C camera, the lens barrel needed to fit the mount is just large enough, after everything else that is needed is squeezed in there, for a front element that results in an 18mm f/3.5 entrance pupil as well as a 55mm f/5.6 entrance pupil. The same is the case with 70-300mm f/4-4.6 telephoto lenses. The front elements needed for such lenses can just fit in a lens with a barrel slightly larger than the throat diameters/flange rings of the most common interchangeable systems.

As a side question, the design of both lenses has to be totally different? Or is it just a matter of changing one or two glass elements at the base of the lens to adjust the projected image size?

Well, they don't have to be totally different. But a lens of the same focal length that projects a larger image circle than another lens of the same focal length must also take in a wider angle of view at the front of the lens. FF sensors have a diagonal about 43mm in length. APS-C sensors have a diagonal about 28mm in length. The middle 28mm of the image circle projected by a FF lens will be the same field of view as the full 28mm image circle projected by an APS-C lens with the same focal length. The parts of the image circle projected by the FF lens outside of the middle 28mm will be parts of the scene not captured by the front of the APS-C lens. This requires a larger front element for wider angle lenses. There's not much difference for telephoto lenses where the angles of view are much narrower and front element size is determined by the entrance pupil size needed for a particular f-number at that focal length.

This is why you see a lot of APS-C only narrow angle zoom lenses and very few telephoto APS-C only lenses. The ones you do see are generally 55-200mm or 55-250mm. These APS-C only lenses are able to provide the same angle of view for an APS-C sensor at 200-250mm that requires a 300-400mm lens for a FF camera. The size and weight savings there are based on the less powerful focal length needed to get the same field of view.

But if the manufacturer can make such minimal changes on the lenses avoiding those downsides, why I have never seen they release two versions of the same lens at the same time, one for APS-C and another for full-frame, like they do for different mounts?

For the most part, lenses released by manufacturers are not breaking very much new ground with regard to angles of view and maximum apertures. Instead, they're further refinements of earlier lens designs. To someone unfamiliar with past lens offerings and the different focal lengths needed to provide the same FoV on camera systems with differently sized sensors, the corresponding lens offerings for each format may not be obvious.

Take the typical 24-70mm f/2.8 zoom offered by most (if not all) major camera makers for their FF cameras. The APS-C "equivalent" in terms of angle of view would be a 17-50mm f/2.8. Quite a few makers offer both 24-70mm f/2.8 FF lenses and 17-50mm f/2.8 APS-C lenses that have very similar designs. Another typical very wide angle FF lens is 16-35mm. The APS-C "equivalent" is a 10-22mm. Plenty of manufacturers offer both such lens options. As the angles get wider the same f-number gets tougher and more expensive to maintain, so the 10-22mm APS-C only lenses tend to have smaller maximum apertures than the fastest 16-35mm lenses. On the other end of the focal length ranges, a 55-200mm APS-C lens gives about the same FoV as a 75/80-300 lens on a FF camera.