Wiles proved the presence of a very rigid structure—not the absence—and the presence of this structure implied FLT via the work of other mathematicians.
If you say that “Wiles proved the Taniyama–Shimura conjecture” (for semistable elliptic curves), then I agree: he’s proved a very important structural result in mathematics.
If you say he proved Fermat’s last theorem, then I’d say he’s proved an important-but-probable lack of structure in mathematics.
So yeah, he proved the existence of structure in one area, and (hence) the absence of structure in another area.
And “to prove Fermat’s last theorem, you have to go via proving the Taniyama–Shimura conjecture”, is, to my mind, strong evidence for “proving lack of structure is hard”.
If you say that “Wiles proved the Taniyama–Shimura conjecture” (for semistable elliptic curves), then I agree: he’s proved a very important structural result in mathematics.
If you say he proved Fermat’s last theorem, then I’d say he’s proved an important-but-probable lack of structure in mathematics.
So yeah, he proved the existence of structure in one area, and (hence) the absence of structure in another area.
And “to prove Fermat’s last theorem, you have to go via proving the Taniyama–Shimura conjecture”, is, to my mind, strong evidence for “proving lack of structure is hard”.