You're close, but not right. The light falling on any given area of the image circle is constant regardless of the format the lens is designed for. Otherwise, cropping a photo would change the exposure, which is obviously nonsensical.

To put it another way, the f/stop is representative of the light at each point regardless of sensor size — not of the total amount spread across the whole covered area of the image circle. (I think everything else you have written is correct, except this point which is causing your confusion.) If you take a printed photograph and tear it in half, each part will have received half of the light of the whole — but that measurement is not relevant to exposure.

On the other hand, bigger sensors do inherently receive more light overall. That's why cameras with bigger sensors have an advantage in having less noise at the same ISO (given roughly equal sensor technology).

But, the lens doesn't matter, because the image circle is irrelevant to your photographs — only the part of it you actually record. (Unless of course you are trying to use a lens with an image circle that doesn't cover your sensor, but that's a different issue.) An f/1.4 lens is always two stops (4x) faster than an f/2.8 lens.