What is the purpose ... for the existing convention?
Math. It's because in many equations regarding simple optics, the ratio N = ƒ/D (where N is the ƒ-number, and D is the lens (or more often precisely, entrance pupil) diameter) pops up a lot, or the use of the ratio simplifies the expression or understanding of the expression.
Example 1: The hyperfocal distance H is the focal distance that theoretically maximizes total depth of field. For a lens of focal length ƒ set to a ƒ-number N, then given a circle of confusion limit c, the hyperfocal distance is defined as
H = (ƒ²/Nc) + ƒ
= ƒ∙(ƒ/Nc + 1)
= ƒ∙(D/c + 1) ≈ ƒ D/c (because D ≫ c)
The formula for hyperfocal distance is merely a special case of the computation of far depth-of-field when the far focus distance is infinity. The geometry that describes the depth of field equations is completely described by similar right-triangles in the cross-sectional plane through the optical axis of the lens, and the thin-lens equation relating the focal length (strength) of the lens and its object-side and image-side focus distances.
Now while the presence of ƒ² in the first hyperfocal distance equation (that includes N in the denominator) might appear to be a result of some dependence on area, it's really just an artificial creation because of the simple algebraic substitution N=ƒ/D. In other words, as long as the aperture diameter D is much larger than the circle of confusion diameter c, the hyperfocal distance is linearly proportional to both ƒ and D, and inversely proportional to c. The equation has nothing to do with the area of the aperture that would be generated by rotating the cross-section of the thin lens of diameter D through 𝜋 radians.
Example 2: A flash's guide-number GN is the product of the flash-to-subject-distance s and ƒ-number:
GN = N ∙ s
Interestingly, the guide number concept is derived from an area relation (which at first glance would seem to support the premise of your question, but as we'll see, there's no need to use square factors). The amount of light incident on an object is inversely proportional to the square of the distance between the light source and the object (the inverse-square law): I ∝ 1/s².
For a given intensity I on the subject, we set our camera's exposure settings to correctly expose the subject. Because we're talking about flash photography, let's assume ISO and shutter speed aren't really free variables available to us for exposure control (e.g., let's leave ISO fixed at 100, and shutter speed at, say, 1/200). That leaves aperture available for adjustment for correct exposure of the object.
If the distance were changed by a factor of k, then the light intensity falls by k². In order to keep the photometric exposure the same, we need to compensate by increasing the aperture area by k², or the aperture diameter by a factor of k. Thus, for constant exposure, the ratio of flash-subject distance to aperture diameter needs to remain constant.
The guide number encapsulates this dependency. Because ƒ-number N is inversely proportional to aperture diameter, the constant exposure relation is now a product rather than a ratio: N ∙ s. And importantly, the dependence on squares of distances is not necessary. We can just use linear flash-to-subject distance and linear aperture diameter.
Regarding units and dimension: Note that N is a unitless quantity, defined as the ratio of two distance-measures (i.e., millimeters divided by millimeters) that are implicitly understood to be arranged at right angles to each other. If N were instead a ratio of focal length to entrance pupil area, the units of N would be in length-1, such as "per meter" or "per millimeter". Net exponents of distance in the denominator is a particularly unwieldy thing for humans to think about and get their head around, in physical models.
Also, having unbalanced ratios of distances would pin the number to choice of units. Any values of such an area-based ƒ-number would be explicitly dependent upon choice of units used for focal length. So aperture settings on lenses with fractional-inch based focal lengths would have completely different values than for millimeter-value focal length lenses (and also for centimeter-valued focal length lenses).
Normalization with respect to fundamental "figures of merit" happens all the time. The first thing to pop to my mind is in relativistic physics. We talk all the time about velocities as some fraction of the speed of light, c, which is approximately 3 x 108 m/s, or about 186,282 mi/s. We don't talk in absolute values of meters per second or miles per second. But in terms of fractions of c, it's much more useful.
Perhaps a better analogy for argument is the debate over what is the better circle constant, 𝜏 ≈ 6.28 vs. 𝜋 ≈ 3.14 (Tau Manifesto). The debate is really a non-debate—as long as the correct factor of 2 is used in the right places, it doesn't matter. One notation might lead to a better understanding of the geometry or physics being described by the equations, but in the end, the math doesn't change. Just the notation and more or fewer factors of 2. Just like aperture diameter vs. area.