Wikipedia contradicts itself in a few points there anyway. Disregard all parts where they talk about equivalent, its bullshit. Basically just made up by the camera industry for common consumers who dont know what an angle is.
Furthermore, a camera with a wide aperture (diameter:distance ratio) achieved by making the entire optical assembly small in all dimensions will perform very poorly when compared to a geometrically similar unit on a larger scale.
This is where you are wrong. It will simply shring the image cycle but not alter the quality of the projected image.
The interesting part is the opposide, if you try to make a phone lens bigger you will get a rather good lens that would be expensive to produce.
This is why phone lenes are of good quality, they cost nothing to produce in that size.
You are looking at photography from the eyes of a physicist whereas in my head it keeps pondering "what about the image cycle?" and "Its not about getting much light!".
Brightness is indeed just the brightness of the projected image, normally you take only a slice of it for you sensor and throw the rest away. What you propose here is to use all of it, that is called speedboosting. Basically the opposide of enlarging an image, shrink it down to increase the light density.
Yes you can do that, but it is not a normal lens design and i have no idea how the result behaves. Most likely its worse than just using a proper sized sensor or else everyone would do it.
For normal lens designs it simply makes no sense to use a larger apperature because the focal plane gets too thin. Say you create a phone lens with f/1 then you can shoot the tip of you nose but not the eyes behind it, let alone that you have a hard time to even focus on said tip.
Or in other words, sure you can get more light by just removing everything and expose the sensor directly, but that wont make a picture.