I am confused by your negative reaction to a putative assertion that every Galois repn's L-function is automorphic (perhaps meaning that it is the standard L-function attached to a cuspidal automorphic form on some $GL_n$ over $\mathbb Q$, and, thus, has the expected analytic continuation and functional equation). To me, various forms of the assertion that every motivic L-function is automorphic (with what is implied...) is one of the best capsulizations of "L's conjectures".
Ok, yes, this does mostly disregard the obvious intuitive senses of "functoriality", refering to putative maps/correspondences of afms on one group to another. And, yes, we know (potential modularity etc: Harris-Taylor's Sato-Tate, et alia) that just a lil' bit of "modularity" goes a long way...
I think that combining the "raw" conjectural ideas with the other dose of conjecture, namely, about existence of a group whose tensor/Tannakian/whatever category is ... automorphic forms...?... may add enough ambiguity to leave it all tooo amgiguousambiguous. Or, perhaps, those things will provoke someone's imagination?
But, seriously, two operational components come to mind: motivic L-functions are automorphic, and, "functoriality" is valid for automorphic L-functions.