to bound roots of a polynomial in terms of its coefficients is in general extremely difficult.

The intuitive difference between this and Ihara zeta is that Ihara involves reciprocal of a characteristic polynomial and hence by Cauchy identity is related to $\mathrm{Sym}^k A$ whereas $\mathrm{Tr} \bigwedge^k A$ shows up in the characteristic polynomial. So closed paths are telling you something about the coefficients of the characteristic polynomial. But there is also a formula for log of characteristic polynomial (in some region of parameter) in terms of a power series involving all the $\mathrm{Tr} (A^k)$. I don't know if any of this is helpful... of course trying...

More precisely, $\mathrm{Tr}\bigwedge^k A$ has a term for each vertex disjoint collection of closed (oriented) paths (as above) whose lengths sum to $k$ (every term comes from a permutation in $\mathrm{Sym}_k$ and the cycle type of this permutation dictates the lengths of the paths). The term also contains the sign of the permutation it comes from: this probably makes it useless for analytic purposes unless you know there are no closed paths of certain lengths.

I added some more information: after thinking about it I realized it is not exactly trails that show up but a weaker variant: I added the details anyway.