Does the functional equation of the Selberg Zeta function imply the Selberg trace formula?
BTW, the trace formula implies the functional equation.
No, you also need that there aren't too many zeros.
Then it works. I indicated the strategy here The Guinand-Weil explicit formula without entire function theory just in terms of the Riemann Zeta function.
Also there is a Lecture Notes in Mathematics by Jürgen Fischer (PhD thesis).