Of course, there's Serre's *Analogues kählériens de certaines conjectures de Weil*, Annals 1960, where he deduces an analogue of the Weil-Riemann hypothesis over $\mathbb{C}$ using standard facts from Hodge theory.
This is technically not an answer at all, but I thought I'd mention it since I had the
(perhaps mistaken) impression that this was partly the inspiration for the standard conjectures.

The other more relevant comment is that one can give an elementary proof of the Weil conjecture for any smooth variety whose Grothendieck motive lies in the tensor
category generated by curves. I should explain,
especially in light of Minhyong's comments, that this could be understood as shorthand
for saying the variety can built up from curves by taking products, taking images, blow ups along centres which of the same type, and so on. Actually, for such varieties, the
Frobenius can be seen to act semisimply. I think this open in general. So perhaps there's
some value in this.