It may have been stated earlier, but here's where I think your confusion may be at:
A larger sensor takes in a greater amount of light because it's larger. The larger your format, the larger your lenses get. So, a fast aperture lens for full frame might be a bit larger than a fast aperture lens for a crop sensor camera. Letting in more light doesn't inherently mean more detail, more dynamic range, etc, etc. Simplified, aperture is all you really need to look at for your practical applications: An f1.8 lens that operates on FX, when put on DX, is still an f1.8 lens. It lets the same amount of light in.
Exactly right, but simply to provide easy examples:
Larger sensors do "collect more light" in the same sense that it takes more paint to paint a barn than to paint one board. More paint to cover the larger area.
But the color is the same either way (which is exposure, illumination intensity per square area).
Same way with f/stop.
Comparing a 200mm lens with a 50 mm lens:
The 200 mm lens magnifies the image, so (compared to a 50mm lens), (and standing in the same place), if we could see the same field of view, the larger 200mm lens image necessarily covers 4x sensor dimensions, which is 16x area (as compared to the 50 mm lens image). This requires 16x more light to cover the larger field with the same previous intensity (same light per square area).
Or (what we actually do), to fit both fields of view onto the same sensor area, the 200 mm lens has to stand back 4x farther (to see same view). This concentrates the same subject scene into a 1/4 size area, which is also 1/16 intensity.
So, to achieve the same exposure (in either example), the 200mm lens aperture must necessarily be 4x larger diameter, which is then the same 16x more intensity (Pi r squared, area is exposure).
This is what people are arguing, but it has all already been solved many years ago. Solution is called f/stops.
We know to instead beneficially use the f/stop numbers to judge exposures...
fstop = focal length / effective lens aperture. This equalizes the lenses, so that any lens at f/4 exposes at f/4. This is the intended single purpose of the f/stop numbering system.
I say effective diameter, because this is the apparent diameter seen when looking through the magnification of the front lens elements (the apparent size the scene sees). The physical diameter will be a bit different.