I think this one is really for the historians. I would guess that the thinking goes back to Emil Artin, and would have been developed in his lectures. Some of that can be seen in the polished edition "Algebraic Numbers and Algebraic Functions", called second edition (1967), but we can assume the ideas are from 20 years before. Artin's ideas can be seen in Lang, "Algebraic Number Theory" also, and (on different topics such as the Newton polygon) in Guido Weiss.
In other words, this is one "school of thought". The connection with the sheaf-theoretical view was probably worked out in greatest detail by Iwasawa (1950s). The Chevalley-Weil school of thought, which after all initiated the techniques, had different ideas on exposition, and the material surfaced in Weil, "Basic Number Theory".
Exposition and motivation is where this all belongs. The "global field" concept is a power powerful set of analogies for those coming from number theory. For those coming from geometry, the fact that there are multiple ways of dealing with one dimensional Riemann-Roch is less suggestive, given that the issue for about 60 years has been about all dimensions.