> ... I find it unsatisfying that the solution to such a
  > simply stated theorem requires 100s of pages of mathematics.

The mathematics involved is getting easier and easier to understand as more and more work is being put into it. Although Wiles originally only proved the underlying result for semi-stable elliptic curves (which was enough for FLT), the full modularity theorem (formerly called the Taniyama–Shimura–Weil conjecture) is now proven, and the field continues to grow. I have little doubt that in 100 years or so this will be within the scope of later stage undergraduate work, just as many results in number theory (and other areas) become more and more accessible as we come to understand them properly.

Wiles' proof is still comparatively new, it's still messy, it needs to be enhanced, clarified, expanded, and then "re-factored" to find the cleanest path to the result. The underlying arc of the proof is relatively easy to understand, it's the details that need the long, tedious, and careful arguments that take 100s of pages. One day it will be an obvious consequence of the full Modularity Theorem, discussed in honors projects of undergraduates.

  > Would have been cooler if the proof was extremely simple to
  > write out, but required outside-the-box thought process that
  > evaded thousands of humans for centuries.

In some sense, perhaps. That's what happened with the proof that primality testing is in P. Three undergrads produced an elementary argument that had eluded people for centuries. And yet it's provided no really new insights into anything, and seems to be effectively a dead end. In that sense it's much more exciting to have genuinely new work that wasn't just a simple argument overlooked for centuries.