The Hilbert–Pólya conjecture is the name given to the idea that the "reason" or "explanation" for the collinearity of the non-trivial zeros of the Riemann zeta function $\zeta(s)$ is that they are the spectrum of some self-adjoint operator. The empirical evidence for the Montgomery pair-correlation conjecture provides some support for the idea that the zeros of $\zeta(s)$ are "spectral."

In the function field case, the Riemann hypothesis has been famously proved by Deligne. My question is, is the intuition behind the Hilbert–Pólya conjecture vindicated in the function-field case?

As I understand it, in the function field case, there is an interpretation of the Riemann zeros as the zeros of a linear operator (a Frobenius action on cohomology). However, it's not clear to me that this necessarily answers my question affirmatively. It seems to me that for the (original) Riemann hypothesis to be "explained" by the spectrum of a self-adjoint operator, it isn't enough to demonstrate the mere existence of such an operator; there should be some "natural reason" for the operator to be self-adjoint. To put it another way, suppose someone were to prove the Riemann hypothesis by some kind of seemingly ad hoc computation that just happened to come out right, and then after the fact, we were to artificially construct a self-adjoint operator to fit the zeros. This would prove the Hilbert–Pólya conjecture in some sense, but it would seem not to satisfy the original desire to "explain" the Riemann hypothesis spectrally.

In the function field case, does Deligne's proof seem to give a satisfying "spectral explanation" of the Riemann hypothesis, or does it feel more like a calculation that just happens to work out?

I realize that this is a vague question that may be largely opinion-based, but I'm holding out hope that people with a thorough understanding of Deligne's proof will find this to be a meaningful question.