So, why can't the sensor's image data downloaded to the processor, globally? Why is it downloaded row by row?

It's a matter of physical limitations and simplicity. The physical limitation is that there's only space for a certain number of external connections -- you couldn't possibly connect every pixel to the processor and grab all that data at once because there's not enough room for the 20 million (or even more) little wires that would have to go from the sensor to the processor. There's not even enough room (on either the sensor or the processor) for enough external connections to read all the pixels in a single row at exactly the same time. So, you need to be able to store the image on the sensor somehow so that you can read it back a little bit at a time.

What you could do is have each pixel connect to its own little memory cell, so that when you trigger the "shutter" every pixel would store its value in its respective memory location at exactly the same time. That's called a global shutter, and some image sensors work exactly that way. But that adds complexity, which means that more chips out of every batch will be defective, which means lower yield from each silicon wafer, which means more expensive sensors. Also, all that extra circuitry takes up space that could better be devoted to more or larger pixels. So what they do instead is to build in enough memory cells to image a single row all at once. That row is sent to the processor and then the next row is read, and so on. This is the rolling shutter.