Why doesn't the picture become darker the more you zoom in?
If the entrance pupil size remains constant, it does.
But very few zoom lenses, even those with variable maximum apertures, maintain the same entrance pupil size as the lens is zoomed.
As your lens's focal length gets longer, fewer photons pass through the lens to hit the mirror/sensor.
Again, only if the entrance pupil size remains constant.
But to maintain the same f-number, the diameter of the entrance pupil is required to scale up at the same rate as the focal length. If you double the focal length you must also double the diameter of the entrance pupil, which quadruples the area of the e.p., to maintain the same f-number.
The physical size of the diaphragm is only part of what determines the maximum aperture, expressed as an f-number, of a lens. Magnification between the front of the lens and the location of the diaphragm also plays a part. The f-number of an aperture is determined by the ratio of the lens' focal length divided by the diameter of the entrance pupil, often referred to as the effective aperture.
In simple language, the entrance pupil diameter is defined by how wide the opening of the diaphragm appears when viewed through the front of the lens.
In your example, a 14mm lens with a 114° angle of view has a 5mm wide entrance pupil at f/2.8. For DSLRs and even most mirrorless cameras, a 14mm lens is what is called a retrofocus design. It's more or less the equivalent of a telephoto lens turned around backwards. So the 'magnification' between the aperture diaphragm and the front of the lens is negative. That is, the entrance pupil appears smaller than the actual size of the physical diaphragm! On the other hand, a 90mm lens with a 27° angle of view requires an entrance pupil 32mm in diameter for f/2.8. That's 6.4X wider, or 41X more area than the 5mm entrance pupil of the 14mm lens at f/2.8.
When constant aperture zoom lenses are moved to change the focal length, the magnification between the front of the lens and the diaphragm is what normally changes, not the physical size of the diaphragm. This change in magnification is what allows the entrance pupil to appear larger at longer focal lengths and smaller at shorter focal lengths for the same physical diaphragm. A 70-200mm f/2.8 lens has an entrance pupil 25mm in diameter at 70mm and f/2.8. At 200mm the entrance pupil at f/2.8 is a tad over 71mm wide. The actual physical diaphragm is the same size in both cases. What has changed is the amount of magnification between the diaphragm assembly and the front of the lens.
Note that this same principle is usually in play with variable aperture zoom lenses as well. Take, for example, an 18-300mm f/3.5-5.6 zoom lens. At 18mm the entrance pupil for f/3.5 is roughly 5.14mm wide. At 300mm the entrance pupil for f/5.6 is over ten times that at 53.6mm wide. Notice that most zoom lenses that max out at 300mm and f/5.6 have front elements that are slightly larger than 54mm in diameter. The needed entrance pupil size is the reason! If the entrance pupil at 300mm were still 5.14mm wide as it is at 18mm and f/3.5, the maximum aperture at 300mm would be f/58!
So why don't all zoom lenses use enough magnification to remain at constant aperture throughout the entire zoom range? Primarily the cost associated with the additional size, weight, and complexity needed to produce a constant aperture lens.