The words ‘pixel’ and ‘photosite’ often get a little confused. The former is a part of an image, the latter a part of a sensor.
“Noise also increases as pixel size decreases“
Noise in the captured data increases as photosite size decreases. We’re talking about images coming from one sensor here. The fact that it’s a small sensor is totally immaterial. When we’re comparing full-resolution images with reduced-resolution images, the photosite size is the same.
If you average down the pixels to a lower resolution then you’re averaging out the random noise, and the noise (in the resulting image) will thus reduce.
This is actually in agreement with the statement you quote, since if you downsize to 1/4 of the original resolution (ie halve both width and height of the image) a pixel in the final image now represents four times the physical area – ie you have larger pixels, thus less noise.