So the paper fails at proving the the second incompleteness theorem, or is it just claiming a bit too much beside the theorem? Is it easily salvageable?

Mmh, that's a nice argument. I assume for vertices $x,y$, $xy$ means the distance from $x$ to $y$ ? Also, is that a standard kind of argument ?

@YCor: I made the question explicit. I hope this is precise enough now.

@AlexandreEremenko: Perfect, I'll look into that. Thanks a lot!

@AlexandreEremenko: OK, thanks! any place where I could look up for such a proof then? I don't have a good intuition on this kind of things.

I'm confused by the following: you say “You can prove a weak form of Cauchy theorem which says that if a function is analytic in $|z|<R≤∞$ then its Taylor series has radius of convergence at least $R$.”, and link to the “Liouville's theorem with your bare hands” post for this fact, but the linked answer only shows Liouville, assuming the Taylor series has infinite convergence radius. Isn't the actual "analytic on $R$-disc implies convergence radius $\geq R$" part actually missing?

Well, I think this answers my question then!