The point is that if there were a small graph that was a counter-example, then there would be large graph counter-examples (and the percentage of them would increase with graph size), so proving that the 4 color theorem is true for large graphs implies that it is true for all graph sizes.

You're claiming:

> the argument looks like it's based on large n asymptotics, so even assuming everything works correctly the strongest statement they can hope to show is that the theorem is true for all n > n_0, where n_0 is some large constant. but there is no mention of this fact. the theorem is claimed for all n.

They're claiming:

> Theorem 7. If there is a map L which cannot be 4-coloured then only an exponentially small fraction of the maps with n edges can be 4-coloured.

These claims appear mutually exclusive.