lens - Why does crop factor apply with APS-C lenses, and why aren't these brighter than FF ones at same aperture?

I know that the image circle of FF-lenses is bigger than it needs to be on APS-C sensors, thus the FOV spears narrower. I've heard that APS-C lenses sit closer to the sensor, thus projecting a smaller image circle around it. To this, I have two questions:

1. 
   

   Why does crop factor apply with APS-C-lenses, while it sounds like the image circle is compressed onto the APS-C-sensor (thus making a wider FOV)?

   
   



2. 
   

   Because of the assumed light-compression, why aren't APS-C-lenses brighter on APS-C-sensors than FF-lenses at the same apertures?

   
   

I assume there is one answer to both questions. I have asked my friend google, but he couldn't come up with an explanation.

Answer

Your logic is sound. If your assumptions were right, then your conclusion would be right.

Let me turn one of your questions around. You ask:

Why does crop factor apply with APS-C-lenses, while it sounds like the image circle is compressed onto the APS-C-sensor (thus making a wider FOV)?

In fact, the image circle isn't compressed, and does not make a wider FOV. It just doesn't extend as far outside of the frame as the circle projected by a lens designed for a larger format would. So the first part is naturally true: the actual projected image within that circle is the same for any focal length, and so if you take less of it, you're cropping — or, "the crop factor applies".

Because of the assumed light-compression, why aren't APS-C-lenses brighter on APS-C-sensors than FF-lenses at the same apertures?

Again, because there isn't any. So, remove that bad assumption and replace it with "the image circle is a design parameter not directly related to sensor size". To quote again:

I know that the image circle of FF-lenses is bigger than it needs to be on APS-C sensors, thus the FOV appears narrower.

This is not true. The FOV appears narrower only because the smaller sensor picks up less of the image circle, regardless of how big that image circle is. There's more on this at Do the same camera settings lead to the same exposure across different sensor sizes?.

However, there are lenses adapters that do work basically this way: "speedboosters" (see How can a speedbooster improve the light performance of a lens?). These do "compress the light" to a smaller circle. But note that by doing so, they also change the focal length. When you calculate the exposure per area of the result, taking into account the new focal length and the new effective aperture, it will be no different from a full-frame (or large-format!) lens of that same effective aperture.