I am not sure I understand your question. Or rather, as I understand your question, you have already answered it. Well, you asked "what would we loose if we..." and you answered "Wiles' proof of FLT". Isn't that enough? We have developed more complicated theories for much less venerable questions!
Admitting rhetorically that it isn't enough, one can develop your answer. If we want to restrict our consideration to representation that are a priori modular, we would loose not only Wiles' proof of FLT, but also Wiles and others's proof of the Taniyama-Shimura-Weil conjecture (and this one has nothing to do with the Frey's curve), so we would lose the corollary that L-functions of elliptic curves satisfy a functional equation and have an analytic extension... We would then not even be able to formulate the beautiful conjecture of Birch and Swinnerton-Dyer! We would also loose that the symmetric powers of the representation attached to an elliptic curve are modular. So we would lose the Sato-Tate conjecture as well.
In a word, we would loose all the Galois representations coming from algebraic geometry, because we don't know a priori that they are modular/automorphic. And the fact is that those representations contain a lot of deep arithmetic and geometric (if you believe to Tate's conjecture) information.
Added: The above is about the fact that without talking of representations that are not modular/automorphic we would lose the ability even to formulate a theorem of the form "such and such representation is modular". Here's another, different, point. We also need Galois representations that are not modular, in order to prove things about representations that are modular. I am refereeing to all the arguments of p-adic families (Hida's family, eigencurve, eigenvarieties) who carry at each point a Galois representation which most of the time is not modular. but we need those point to give flesh to the eigenvariety whose "skeleton" is made of the modular ones. To give an early example of those techniques, Wiles proved that a Galois representation attached to an ordinary form is ordinary (a statement purely about modular galois representation) using Hida's families, hence non-modular representations. Nowadays those techniques are ubiquitous...