After years of watching numberphile on Youtube, my guess would be that there's a cutoff number n0 for which can be inductively shown that if n0 is a period then n0+1 is also a period; that would then eliminate almost all natural numbers from the search. Such a proof might perhaps use combining periodic patterns to obtain longer periods. It seems to be difficult to construct a periodic GoL pattern with a given period, so that then would explain why of all numbers two very small ones held out over the decades; this is truly astounding, especially given how incredibly inventive and industrious people have gotten with the Game of Life, simulating entire computers on the canvas.