Generally speaking, a larger front element is necessary to achieve a wider maximum aperture. More specifically, a larger front element helps achieve the necessary "entrance pupil" diameter required for a given lens, provides the necessary primary light-gathering power of a lens, and helps achieve the necessary angle of view of the lens. (The entrance pupil is the diameter of the physical aperture as viewed through the front of the lens.)

The physical diameter of a lens generally must increase as the maximum aperture increases, and once you pass f/2.8, each additional stop greatly increases the physical size of the lens. Additionally, once you pass f/2.8, each additional stop requires a considerably greater amount of light, and larger front lens elements are a key factor in gathering that additional light.

For ultra-wide angle lenses, such as the 14mm f/2.8, a larger lens element is often necessary to assist in capturing light rays from a wide enough angle of incidence, more so than for achieving a wide aperture (14/2.8 = 5mm physical aperture, quite small.)

For wider-aperture telephoto lenses, the physical aperture tends to be much larger, which tends to dictate the size of the front lens element more than the necessity of gathering wide-angle incident light rays. The 70-200mm f/2.8 lenses have a physical aperture of 71.4mm, some 14 times larger than the 14mm f/2.8 lens.

Lenses like the 70-300f/4.5-5.6 and 24-120 f/3.5-5.6 have much smaller maximum apertures for their focal lengths. 300/5.6 = 53mm, some 1.5 times smaller for 100mm greater focal length. A 300mm f/2.8 lens would require a 107mm aperture, which is twice as large as a 300 f/5.6, and would require a much larger front lens element to gather enough light to accommodate such a large aperture. The 80-400mm again has a fairly small maximum aperture at its longest focal length...400/5.6 is 71.4mm again, vs. 100mm for the 200/2 and 107mm for the 300/2.8. The 80-400mm lens has a larger front element than say the 14/2.8 or even a 50/1.4 due to the physical size of its aperture...which even at f/5.6 is considerably larger than any wide angle lens. A 50mm f/1.0 lens would have a physical aperture of 50mm, which over 20mm smaller than the 71.4mm of a 400/5.6 lens.