Why is it so important to have the phase gradient encoding step come before the frequency gradient encoding step when collecting MR Images? Currently, the phase encoding step is turned on briefly, then the frequency encoding step is turned on during data acquisition, effectively populating k-space row by row. Could you reverse the order of these steps?
You could reverse the order - you could switch the frequency encoding and the phase encoding axis. This would then populate $k$-space e.g. not from "left to right", but from "top to bottom". However, the phase encoding step would always come before the frequency encoding step.
In MRI, the gradient can be thought of as the speed that you travel in with in $k$-space. Since in cartesian encoding that you asked for, you have two perpendicular directions (and data is filled in a simple 2D array), you need to travel with a constant speed (otherwise you would have different step sizes in $k$-space and could not sort the data into a 2D array without further processing, e.g. regridding).