This is both an answer and a question:

As part of a response to a previous question of mine, David Speyer wrote that:

... it is known how to adapt Weil's proof of the Riemann hypothesis to higher dimensional S, if one had an analogue of the Hodge index theorem for $S \times S$ in characteristic p. I've been told that a good reference for this is Kleiman's Algebraic Cycles and the Weil Conjectures...

So perhaps a "moral" proof would require a Hodge index theorem in characteristic p.

However, David later writes that Grothendieck's standard conjectures assert that the Hodge theorem holds. So is this possible proof the same as "Grothendieck's envisaged" one?

This is both an answer and a question:

As part of a response to a previous question of mine, David Speyer wrote that:

... it is known how to adapt Weil's proof of the Riemann hypothesis to higher dimensional S, if one had an analogue of the Hodge index theorem for $S \times S$ in characteristic p. I've been told that a good reference for this is Kleiman's Algebraic Cycles and the Weil Conjectures...

So perhaps a "moral" proof would require a Hodge index theorem in characteristic p.

However, David later writes that Grothendieck's standard conjectures assert that the Hodge theorem holds. So is this possible proof the same as "Grothendieck's envisaged" one?