Then please correct me.
If you have the same amount of light falling on the sensor, there's no difference no matter how large the sensor is (given that they have the same amount of pixels), the same amount of light will still hit the sensor. Each pixels will still get the same amount of photons, only that the smaller sensors have much denser photons than the larger sensor.
Correct me if you wish, cause' I don't like to see myself being misinformed for too long.
There are two problems with your reasoning. The first is that the same AMOUNT of light does not fall on a ASP-C sensor as a FF sensor given the same lens. The same DENSITY of light does, but the larger area of the FF sensor allows it to collect more light.
The second problem with your reasoning is that the density of light doesn't change, meaning it also doesn't vary depending on the number of photosites. Consider the case of your sensor only having one pixel. That one pixel will collect all of the light. Now consider your sensor having two pixels. Each pixel will collect half the light. This is because each pixel will occupy half the area and the light hitting each pixel maintains the same density. As a whole, the 2 pixel sensor collects the same amount of light as the one pixel sensor because the total area of both sensors is the same. Your statement "Each pixels will still get the same amount of photons, only that the smaller sensors have much denser photons than the larger sensor" cannot be true because a sensor with more pixels AND denser light would mean more light had to have been input. It's not more dense and it's not less dense... it is the same density over a smaller area, hence less light collected by each photosite.
Ah, you're essentially saying what I'm trying to say. What I mean from an APS-C lens is a lens that only covers the APS-C sensor, not a full frame lens that covers for full frame but used on APS-C sensor.
Do you realise that even at the same f/stop, full frame lenses are larger than APS-C lenses? If you made the APS-C lens the same size as the full frame lens, then the same total amount light will hit the sensor. Density of light does change. A larger f/stop will bring higher density of light, even if the total amount of light is the same. The reason DX *seems* to have worse low light capabilities is because most of the time, when people compare DX with FX, they're using FX lens and the same aperture. The FX lens is not made for the DX sensor, so, you'll be wasting a lot of the glass. If you utilise the entire glass, a DX lens at the same size as FX will essentially transmit about the same amount of light to the sensor.
Your statement "Each pixels will still get the same amount of photons, only that the smaller sensors have much denser photons than the larger sensor" cannot be true because a sensor with more pixels AND denser light would mean more light had to have been input.
That is simply not true. Here is an easier explanation. Humans are pixels on full frame, mice are pixels on cropped frame, apples are photon, fullness is density of photon. There is 100 apples given to 100 humans and 100 apples given to 100 mice. Each human and mouse gets to eat 1 apple. To a human, they don't feel really full. To a mice, they feel really full, even though they actually got the same apples.
Because light are more spread out in full frame, they're less dense. DX sensor size is smaller light are not as spread out, so the light is denser.
Another way to think of this - at the same weight, short people are normally fatter/larger/denser than tall people.
Although I might not be sure if full frame really had no low light advantage compared to cropped frame, I'm pretty sure if both full frame and cropped frame get the same total amount of light (same amount of light per pixel), cropped frame will always get denser light.