Some counterpoints: 2 mathematicians greatly increases chances of senility. Blogs/twitter also correlate much stronger with author age than truth of statement.

>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.

This is completely wrong. A proof for all n>n_0 is a proof for all n, since any counter-examples have to exist as subgraphs of arbitrarily large graphs.