Now that I've been exposed for a little longer to literature about Langlands (this answer is a month after I asked this question), let me submit my own non-expert answer to this question. I did enjoy the generality and naturality of the formulation of Langlands described in a newer question of mine:

How does the conjectural Langlands group fit into the Tannakian point of view?

As you can see from that question, I'm still learning about this. But I humbly suggest that this formulation might be what I was looking for, and so for the sake of completeness I add this as an answer.

Now that I've been exposed for a little longer to literature about Langlands (this answer is a month after I asked this question), let me submit my own non-expert answer to this question. I did enjoy the generality and naturality of the formulation of Langlands described in a newer question of mine:

How does the conjectural Langlands group fit into the Tannakian point of view?

As you can see from that question, I'm still learning about this. But I humbly suggest that this formulation might be what I was looking for, and so for the sake of completeness I add this as an answer.