If I could place the RX100 sensor in A500 for bird shooting, .........

It depends on what you wanna use the aperture for. In the end you can do everything without the FF equivalent aperture but you cannot without the real aperture.

If you care about the shutter speed, all that matters is the real aperture.

If you care about the DOF you do not need crop factor or relative aperture. Just use the focal length, the real aperture and the size of the sensor. No need for the fancy relative aperture.

If you care about composing the picture and the field of view you do not need the relative aperture either. Just look through the viewfinder or calculate the stuff (without the real aperture).

The relative aperture only matters when you want to compare FF aperture to different sensor sizes. And even there you can just calculate everything that matters without even knowing the crop factor or relative aperture.

Light gathering abilities do not matter in any way. They do not directly influence every day photography. Using this as an argument does not make sense.
Every sensor is acting different when getting exposed to the same number of photons. It makes more sense to compare the noise at certain ISO levels and comparing the dynamic range.

All you do is confuse people. A FF lens mounted to APS-C does not change in any way and the different sensor sizes have a certain effect on the resulting picture.

Most of the people never use FF cameras. Even in the rare case someone wanna compare the aperture on a non FF sensor to anything else calculating an aperture ratio using an APS-C sensor as the comparing element would make more sense. Almost everybody used a APS-C camera at some point.