You're missing something. When it comes to photometric exposure, it's the angular area of the light cone entering the lens that matters. As long as their lenses cover the same angle of view, a very large camera and a smartphone camera will have the same photometric exposure. The same amount of photons have the same opportunity to enter the same light cone.

That's exactly why we use the f-number: it eliminates the need to know precise, exact aperture areas. Because the f-number is a ratio of two orthogonal measurements — that is, the lens's focal length and the entrance pupil's diameter are at right angles to each other — it is another way of describing an angle, mapped through the tangent function. As previously mentioned, the same light cone angles from the same camera perspective see the same light intensity, but that's just a tangent function away from saying the same focal-length-to-aperture-diameter ratios (i.e., f-numbers) see the same light intensity.

Note that actual front lens element diameter is not the lens’s aperture. It’s size of the entrance pupil, the aperture when seen from the front of the lens. For telephoto lenses, the front element needs to be large enough to see the entire iris image, or else it would act as a field stop, limiting the maximum aperture size.

But looking at the front of a wide angle lens, you’ll notice the entrance pupil is much smaller than the front element. That’s because the front element is large (and heavily curved and bulbous) in order to gather light from a wider field of view.