Even if Fermat didn't have a proof, many mathematicians hope that a proof will be found that is much briefer than Wiles', which is over a thousand pages. It's unsatisfying in the sense that the theorem is now known to be true, but the proof doesn't provide an understanding of "why" it's true.
I'm not sure where you get the over a thousand pages figure for Wiles' proof, as this is not accurate. The proof credited to Wiles (and a co-author) was published across two papers, "Modular elliptic curves and Fermat's Last Theorem" which is 109 pages and "Ring theoretic properties of certain Hecke algebras" which is 20 pages, for a total of 129 pages.
Furthermore, neither paper was written for the purpose of attempting to provide a direct proof of Fermat's Last Theorem. The papers were written to provide a rigorous closed-form proof to previous conjectures which had, up until then, been supported only through numerical methods. One of the consequences of the proof was its application to solving Fermat's Last Theorem. So, solving FLT was only a small (albeit very interesting and noteworthy) component of the papers.
I would also say it's a mischaracterization to state that the proof doesn't provide an understanding of the "why" involved. If the proof is valid, which it is, then both the how and the why have been rigorously and exhaustively explained. Granted the mathematics involved is not trivial, but I think it is unfair to characterize something as "unsatisfying" or incomplete simply because it is not easily accessible. Many things in science, mathematics, and everyday life are well-understood, despite being incredibly complicated to express in a closed form.
On another note, one of my nit-picks about the episode is the gambling scenes. It is obvious the production team did not consult a casino/gaming expert in how to stage them. The dealing techniques, card handling, payouts, etc. were all very amateurish. Some of the wagers were illegal or nonsensical. And some of Data's strategy advice when it comes to blackjack was incorrect when it comes to maximizing the expected value of the hand. Surely a walking computer could calculate basic probabilities better than he did!
But this episode will always hold a nostalgic place in my heart because it is the very first episode of Star Trek: The Next Generation that I ever saw.