Although this succeeds in eliminating 90% of the search space for a 2x2 grid (and even then I think your sums are slightly wrong) this doesn't even dent the 17x17 case. The 2x2 square in the top left of that only has 23 (or fewer) configurations, but now you can probably no longer permute the colors. The savings are irrelevant in the larger cases.
Besides, to some extent you are preempting the problem. You've counted the number of colorings of the 2x2 grid and found there to be only a small number. The evidence suggests that the number of colorings of the 17x17 grid is either 1 or 0 (up to permutation of colors), but that doesn't mean the search is trivial.
I may not have expressed that clearly, but I hope I've conveyed the point.
There is something like 12, valid 2x2 grids, and you can get rid of almost another (17!)^2 options because rows and columns are interchangeable. Actually thinking about it a little bit the only valid option for the top left square is:
0,0
0,1
However, the vast majority of the grids are eliminated by the boxing constriant before the grid get's all that large.
Besides, to some extent you are preempting the problem. You've counted the number of colorings of the 2x2 grid and found there to be only a small number. The evidence suggests that the number of colorings of the 17x17 grid is either 1 or 0 (up to permutation of colors), but that doesn't mean the search is trivial.
I may not have expressed that clearly, but I hope I've conveyed the point.