~~The only method we know of to prove analytic / meromorphic continuation of zeta-functions of alg. varieties over number fields is to go via some kind of modularity, or potential modularity, statement. So even making sense of the statement of your program ~~**already requires** the key piece of technology developed to prove FLT (and modularity of these high-degree Fermat curves is likely to be *vastly* harder than elliptic curves).

~~
That is, your "whole-hearted belief" makes this proposed program circular.
~~
EDIT: Dan Loughran's comment alerted me to the fact that this is misleading. Fermat curves have lots of automorphisms, and there are enough of these to force the L-function to factor as a product of L-functions of Groessencharacters of cyclotomic fields, for which analytic continuation + functional equation of the L-function are known; see Aoki (1991). This is, of course, an automorphy statement of a kind, but one that's far easier than automorphy of general higher-genus curves would be.

There are several remaining serious obstacles. For instance, once you go beyond genus 1, the link between the special values of the L-function of a curve and the existence or otherwise of rational points on the curve is very indirect. Rather than points on $X$, the $L$-series gives you information about points on $Jac(X)$. Sometimes, with lots of extra work, you can translate this into information about points on $X$ itself (the Chabauty--Coleman method), but there are lots of cases where this does not apply.