Think of the metabones speed booster. Imagine now that they created one which made a FF lens create an imaging circle to match the APS-C crop sensor found in a Canon camera - a 1.6x telecompressor. That way you could mount, say, a 24-70/2.8 on crop with the metabones adapter, and get an identical FoV that the lens achieves on FF.
The 24-70/2.8 would be turned into a 15-44/1.75, right?
Now, we all know f1.75 is faster than f2.8. No-one is disputing that. If you mount this lens on the crop sensor camera with the telecompressor, it will allow for more than a stop faster shutter speeds at equal ISO's. It is an f1.75 lens, and no-one can argue with that. But where does this metabones get this extra speed from? Its not magic - its just it compresses the larger image circle into a smaller one - that extra light from that larger FF image circle is now condensed down into a smaller, more intense imaging circle, and is then received by a smaller sensor. However, in total its only the same amount of light/photons coming in through the lens which hits the sensor. The FF sensor and the APS-C sensor with a telecompressor both receive an identical number of photons, but the APS-C sensor has brighter light presented to it - more light per area - in other words its just over a stop brighter.
I confess this the first time I've heard of a metabones speed booster but I get the idea and, yes, I'm with you so far.
ISO's are rated to make exposure calculations work. What one camera does to achieve ISO 1600 isn't the same as another camera at ISO 1600 - especially with different size formats. The larger sensor as a whole has more photons hitting it at a particular aperture, so it needs to amplify the resulting electrical signal less for any given ISO. And even some cameras with the same sensor have to do different amplification, such as the Sony NEX 7 and Sony SLT A77.
No, you've broken the sequence of logic from your metabones analogy. You can only say that the amount of light hitting the FF sensor is more because the FF sensor is bigger. The amount of light hitting each pixel is exactly the same, regardless of sensor size.
If you can't see that, haven't you ever wondered why FF sensors are typically just over a stop better than crop sensors when it comes to noise? This faster aperture of the Sigma simply allows the noisier sensor to work at lower ISO's to finally fight back. Use a 18-35/1.8 at 35mm f1.8 1/100th of a sec ISO 1000 on crop, or a 24-70/2.8 at 56mm f2.8 1/100th, ISO 1600 on FF and you'll find its the same framing, depth of field, exposure and noise. (well, it would be if the Sigma was slightly brighter at f1.75).
This is a compound error from the previous incorrect statement. Sensor size is not a factor in this.