Not quite an answer, but a heuristics from the point of view of Weil philosophy about why the rationality mod $p$ is much easier.

Weil reduced the rationality of zeta-function to the existence of a good cohomology theory. The crucial property of a cohomology which satisifies the Weil axioms is that it has zero-characteristics coefficients. This is needed because Lefshetz formula is an identity in the coefficients ring, so to compute the number of $\mathbb{F}_q$-points precisely, we need that coefficents ring does not have any $\mathbb{Z}$-torsion.

But if you are interested only in the number of points modulo $p$, a cohomology theory with $\mathbb{F}_p$-coefficients which satisefies the Lefshetz trace foormula would fit(all the formulas for zeta functions should just be reduced modulo $p$).

It is very easy to give an example of such cohomology theory. It is $X\mapsto H^i(X,\mathcal{O}_X)$. Holomorphic Lefshetz trace formula applied to Frobenius gives $$\sum\limits_{x\in X(\mathbb{F}_{p^n})}\frac{1}{\det(1-Fr|_{T_{x,X}})}=\sum\limits_i(-1)^i\mathrm{Tr}\, Fr|_{H^i(X,\mathcal{O}_X)}$$ But Frobenius is inseparable, so its differential is zero and the LHS is stil $\# X(\mathbb{F}_{p^n})$ so we get the desired trace formula modulo $p$.

So, the rationality mod $p$ should be much easier. You can also trace the analogy between Chevalley-Warning method and holomorphic Lefshetz trace formula in the proof of equivalence of different definitions of supersingular elliptic curve(see i.e. Hartshorne).

Historically, the rationality mod $p$ was certainly known already to Serre, since he considered Witt cohomology $H^i(X,W\mathcal{O}_X)$ as a candidate for Weil cohomology.