+1 for the first sentence.

@DanielMoskovich In some sense, even if I had a purely number-theoretic proof of some purely number-theoretic problem I enjoy, I would look for more, because I would feel that the purely number-theoretic proof would not illuminate the role the motivic or automorphic structures play (Dirichlet's theorem above would be a baby example, if you were to find a very clever combinatorial argument, it would be a remarkable achievement, because I think a number of quite clever people tried and failed, but it would not necessarily be a conceptual step forward).

@DanielMoskovich In fact, I think that the statement you ask us to refute and your comment indicates where the crux of the matter lies. You seem to believe that real numbers are tools for number theorists ("We know that calculus works well, so we are tempted to apply it to anything and everything") but to me, in addition to tools, they are primarily part of the very fabric of the phenomenon I study (which is why I'm not surprised that conceptual approaches of these phenomenon requires them heavily).

@DanielMoskovich I don't think I'm saying that (but I'm not sure I understand your question). What I'm saying is that a lot (really a lot) of what is and has historically been classified as interesting number-theoretic phenomenon retrospectively appears to be shadows cast on integers by structures which are only meaningful over complete fields. The "rational-number-based approach to Number Theory" is the study of the shadows, but at some point we learned to escape the Platonic cave and we caught glimpses of the actual structures. I don't expect we are ever going back.

I'm not a graph theorist either, but the statement seemed intuitive to me (it is quite hard for a large random graph not too be connected). Anyway, the article The $k$‐Connectedness of Unlabelled Graph by Walsh and Wright contains a much stronger statement.

"So for large $N$ almost all labeled graphs are connected" Surely that conclusion can be obtained in a much simpler way than with the full asymptotic.