Firstly, I don't think that $\zeta(s)$ is that efficient to compute. Our interest in calculating $\zeta(s)$ for prime number theoretic purposes typically focuses on the critical line $\mathrm{Re}s=1/2$ and the Riemann-Siegel formula requires $O(t^{1/2})$ terms to compute $\zeta(1/2+it)$. There are speed-ups for calculating multiples values, but not dramatically so.
Likewise, I'm not sure what you mean by reverse. The function is not bijective (we know many places where it is zero for example).
That said, there have been some ideas around using analytic number theory for factoring methods. Shanks's class group factorisation method can be sped up if one can approximate $L(1,\chi_N)$ (here the $L$-function is for the number field $\mathbb Q(\sqrt N)$ and is closely related to $\zeta(s)$. Assuming the generalised Riemann hypothesis, Shanks managed to reduce the run time of his algorithm to factor $N$ from $O(N^{1/4+\epsilon})$ to $O(N^{1/5+\epsilon})$. Such complexity is unlikely to factor numbers bigger than a few hundred bits and cannot compete with the general number field sieve.
There have been ideas to use $\zeta(s)$ itself (see the recent paper "Factoring with hints" by Sica for example), but these struggle to get close to the complexity of Shanks's methods of the 1970s (the Sica paper has complexity $O(N^{1/3+\epsilon})$.)
