noise - Is the low-light advantage of larger sensors attributable to the sensor itself or to the larger aperture of the lenses?

Let's say I take a photo using an APS-C camera and using an FF camera, from the same position, with the same ISO and shutter speed.

Let's say I use a 50mm focal length for the FF camera. To match the field of view, I use a 50/1.5 = 33mm focal length for the APS-C camera.

Now, let's say that both lenses have the same aperture of f/2.8. But since the focal length is different (50 vs 33), the actual value of f/2.8 will be higher for the FF lens, which means it's letting in more light [1].

Am I correct in understanding that the FF system performs better not because of the sensor but actually because of the wider opening of lens? If we measured aperture in sq mm, and if I take a photo using the same actual aperture, like 20 square mm, with an FF and an APS-C camera, won't I get the same amount of noise?

In other words, the supposed advantage of the FF system is merely an artefact of the way we measure aperture, as f-numbers. If we chose to measure aperture in sq mm, the advantage of the FF sensor goes away.

I'm assuming we are comparing lenses of FF-equivalent focal length, and we don't have a mismatched system like mounting an APS-C lens on an FF body or vice-versa (unless you use something like the MetaBones Speed Booster).

There are two ways of measuring aperture:

1. 
   

   As an f-number, which is the conventional way of measuring aperture.

   
   


2. 
   

   As a total area, in square mm.

   
   

(1) measures the light per unit area and (2) measures the total light incident on the sensor.

(1) is useful to determine aperture in relation to shutter speed and ISO (the so-called exposure triangle).

(2) is useful when you're interested in noise under low-light. Since noise is caused by lack of light, so you want to measure the total light coming in to the sensor, not the amount of light coming in per unit area of the sensor. (I'm assuming you're viewing both photos at the same size on your monitor.)

Since (2) assumes that you don't have a mismatched system, where you mount an FF lens on an APS-C camera or vice-versa (without something like the MetaBones Speed Booster adapter), and also assumes you're not comparing lenses of different FF-equivalent focal lengths, it may not be a very useful measurement of aperture in photography in general. But it helps to determine noise, taking sensor size out of the question.

Let's say you're trying to buy a camera + lens for low-light work, and you've identified two options within your budget:

1. 
   

   An NEX body with a Sony f/2.8 lens.

   
   


2. 
   
   

   An Olympus µ 4/3 body with an Olympus f/2 or an f/1.4 lens.

   
   

The f-number measurement makes it hard for you to see which system performs better under low-light. Whereas, if we measured aperture in square mm, you'd instantly see which system worked better under low-light -- the one with the bigger aperture in square mm. (Again, let's assume both lenses have the same FF-equivalent focal length.)

A comment below says that it's an academic question whether we attribute the advantage of the FF system to the bigger sensor or to the wider entrance pupil of the lens. The existence of the MetaBones Speed Booster disproves that statement, since you can now mount an FF lens on a crop-factor sensor without losing light or, more formally, without having an image circle larger than the sensor. When you can mix and match FF and APS-C lenses and bodies, it's no longer academic whether you attribute the advantage of the FF system to the body or to the lens.

Answer

(Edited for Compliance:)

The short answer is yes, the way you're looking at the issue of attributing low-light capability to the total light-gathering power of the lens instead of to the area of the sensor does also work to explain the observed differences between different systems. However it is more complicated conceptually and doesn't really provide any advantage over the simpler and more commonly used model of 'bigger sensor = better'.

Everything in photography involves compromises and every system is the result of the combination of its parts. Combining different parts in different ways can achieve the same results, and there can simultaneously be other compromises involved that some people will accept and others will reject. There is never a single correct answer.

The idea that you're on the edges of here has been called "equivalence". More than you'll ever want to read about it is here: josephjamesphotography.com/equivalence/