The subject area is constant, it does not change. The sensor area is constant, it does not change. These are true no matter where the camera is moved.
Yes, but when you move the camera, the image of the subject on the sensor does change size.
The light reflected off of the subject has no idea that it is going to pass through a lens and then onto a sensor. It is more or less intense exactly according to the inverse square law.
Double the subject distance, and you'll tell me the light intensity will be quartered. But double the distance and the area that same light covers on the sensor will also be quartered. A quarter the intensity over a quarter of the area means it is still the same intensity per unit area. That's why exposure doesn't change with subject distance.
But that larger sensor at the closer distance that has the same intensity per unit area, has more area, so it has more light total. It doesn't have more light because of the shorter subject distance. It has more light because it has a larger surface area. You seem to accept that more light results in a better SNR, so, there you have it: Larger sensors collect more light, and thus give better SNR.
Larger sensors have another advantage too. A lens doesn't have to be as sharp to yield a given level of sharpness for a given megapixel count. Or a lens on FF yields a sharper image than that same lens on APS-C for the same portion of the image circle.
Sharpness is essentially a measure of how much of the light from a singel point source falls on the same pixel. When 24 MP are spread over a FF sensor, each pixel covers more than twice the area of a pixel on 24MP "HF" sensor. So a lens only has to be 1/1.5 as sharp on a FF sensor to resole the same as a lens on an APS-C sensor. Or alternately, the same lens mounted on a 24MP FF sensor will result in a sharper image than that lens mounted on a 24MP APS-C sensor, bcaseu it will be able to direct more of th light from a point source onto a larger pixel.
You've talked abour MP count more than once. It has no significant impact on noise level across the image. The 12MP Sony A7s, the 16MP Nikon D4s, the 22 MP Canon 5DIII, the 24MP Nikon D610, and the 36MP Sony A7R all have roughly the same sensor size, and all have nearly identical SNR performance at most ISOs.
In contrast, similar techonolgy APS-C sensors from all three conmpanies have about one stop worse SNR performance at all ISOs. That's pretty much what you'd expect if sensor size was driving noise performance.
The Canon sensor has inferior DR at low ISO. That is because it's older design introduces read noise between the sensor and the ADC. This read noise obscure low level signals, thus reducing DR.