Per my last reply here, by "aperture" I assume you mean "focal ratio". Please read what I said there about misusing that term. What I did not say is that the misuse of that term is part of the conceptual problem many have with this.
Assuming we are talking wide open performance....
Focal ratio is a.... ratio. It is a malleable number (in principle), trading focal length for focal ratio for a given size front element in a telephoto lens. However...
From front element diameter we get area with PiR*2. The area of the front element determines the total quantity of light that hits the sensor. The total number of photons *across the subject* that get counted. It is literally called "Light Gathering Power" (LGP) by those that care about such things. It means what it says. Gooogle it; you will find a better explanation than I can rattle out here tonight.
As I mentioned in my last reply, if you are free to set your working distance however you want, then f/2.8 is f/2.8 as far as noise is concerned.
In the context of a fixed working distance of a fixed size subject, where you want to minimize noise, more Light Gathering Power always wins and is the key to the optimum optical prescription.
The bigger glass allows you to get a longer focal length for a given focal ratio. For example, you want to shoot a bird at f/4 but 500mm is not enough to fill the frame such that you don't have to crop for lack of reach.
You change to a 600/4. Your exposure is the same because you are still f/4. But your bird is 20% bigger so you received 20% (squared) more light across the bird. When you prepare your final output you will magnify the bird 20% less, and therefore you will have less noise. The more you magnify the subject the less noise you get.
You know from your birding that when using high ISO's, it is better to get closer so you need to magnify the bird less. Since you have only one longest lens, you must get closer to do that. But in principle you can stay where you are, and just mount a larger diameter glass, similarly configured to f/4, and it will now have a longer focal length. Better bird picture
Another example is where you are shooting a 400/4 and that is all the reach you need to fill the frame. Going to a 400/2.8 (enlarging the front glass, maintaining focal length and using the excess LGP for faster focal ratio) far more obviously gives you that extra stop.
When you enlarge the glass, then in principle you can re-figure the optics anyway you want, depending on how you want to trade focal length (reach) for focal ratio. In principle. In practice you take what few optical configurations are available and add TC's to reconfigure.
I know this is not necessarily intuitive unless you have studied the implications of Light Gathering Power and lived it, but it is a fact.