I think perhaps the phenomenon you are seeing is someone working on a problem for a long time and not telling the world about it. The better question might be "What makes one wait to announce their particular research program?" That must vary from case to case and reasons might be good or not so good. If you announce a bright idea for attacking a problem it might be ignored or criticized, in which case, why do it? And if it is recognized as a great idea, someone else might beat you to the result.
My recollections of the history of the three you mention are less than perfect, and I didn't recheck. Zhang is a smart guy but was working as an adjunct so may have had reservations about saying anything before he knew he had a complete proof. Wiles was a professor at Princeton (now Oxford) but before his bombshell if you said "I think I can come up with a proof for Fermat's Last Theorem" you would be in for heavy duty scrutiny.
Here is a more typical example. I choose it because I've heard talks by the author. Babai's work on Graph Isomorphism: (follow the link for references etc.)
The best currently accepted theoretical algorithm is due to Babai & Luks (1983). The algorithm has run time $2^{O(\sqrt n \log n)}$ for graphs with n vertices and relies on the classification of finite simple groups.
In November 2015, Babai announced a quasipolynomial time algorithm for all graphs, that is, one with running time $2^{O((\log n)^{c})}$ for some fixed $c>0$. On January 4, 2017, Babai retracted the quasi-polynomial claim and stated a sub-exponential time bound instead after Harald Helfgott discovered a flaw in the proof. On January 9, 2017, Babai announced a correction (published in full on January 19) and restored the quasi-polynomial claim, with Helfgott confirming the fix. Helfgott further claims that one can take $c = 3$, so the running time is $2^{O((\log n)^3)}$. The new proof has not been fully peer-reviewed yet.
I think most in the field were pretty amazed by the result, even though his plan of attack was known and he published results along the way. There are many talks by Babai on the web, here is one. If it is like talks I've heard, he discusses (someplace in those 90 minutes) his wrong turns and breakthroughs over the 30 or so years hammered away at the problem (while doing much else).