There are definitely some results which have this "a bunch of little hacks" quality you're describing, and while opinions differ I share your intuition that there's something a little disappointing about solving a big problem that way.

But I think FLT is about as far as one can get from that situation! Wiles's work was the culmination of centuries of theory-building work, and the concepts that were developed over that time are far more important than FLT; the thing Wiles actually proved (a special case of something called the "Modularity Theorem", the full version of which was proved a bit later) is itself much more valuable to human understanding of mathematics than FLT. It's certainly very cool that it can be used to answer such a simple question that was open for so long, and it makes for a great headline, but I think if you asked number theorists working in the area they would almost all tell you that they're much more grateful for the theory that came out of this quest than for the mere fact that the quest was completed.