Not really any better than the others but a different way to look at it:
Consider two squares in the top row colored A and B. Then the two squares below that are C and D in some order, and below that A and B in some order, then below that C and D, etc. Eventually, in the bottom row, you have a C and D.
What this means is if you remove the whole bottom row and place it on top, the square will still satisfy the key property (of each $2 \times 2$ subsquare having four colors).
By symmetry, you can then remove the whole right side column and move it all the way to the left, and it still satisfies the key property.
Now the squares that use to be in the four corners are located in the first $2 \times 2$ subsquare in the upper left, and thus it is obvious that they are four different colors.