> To stretch our non-mathematical metaphor well beyond its breaking point, a "bigger sphere" will contain more signal as well as more noise, so just expanding the size of your region doesn't affect the SNR. That's why, and I'm very sorry about this, a bigger antenna actually doesn't improve your reception at all.
In my understanding that is incomplete because it ignores amp noise. When your antenna is larger, you get more signal (yes including more noise), so you need to amplify it less before the signal can be sampled. That means less amplification noise and the effective SNR is better. Thus antenna gain, not just antenna directionality, will improve reception.