I think the FCT proof doesn't generate every possible graph in that category (as there are infinitely many of them) but rather enumerates a very large set of compositional planar graphs such that all planar graphs are composed of combinations of subgraphs within the enumerated set
Yes, I think you are correct. I looked at it, and I looked into it, but only to grasp the general gist of it. The point was that essentially they reduced the infinite opportunities for tiling to a finite set of graphs whose composition covered the full spectrum of cases.
