Cantor couldn't prove or disprove the continuum hypothesis which was what drove him crazy according to that documentary (I don't buy that FWIW) not that he proved something absolutely wrong.
I don't think Gödel was extremly shocked by his findings and I'm pretty sure that Turing wasn't shocked at all by his findings given Gödel. I also don't think Gödel's starving issue was causaly related to his most profound insights.
Turing was driven into suicide by his government that had pretty much nothing to do with his work.
I think the better link is "infinity and limitations of knowledge" and that's why I would include Cohen.
Basically Cantor-Gödel-Cohen is the "continuum theme". Gödel proved that the negation of CH cannot be proven ins set theory, Cohen proved that the CH cannot be proven in set theory...mathematicians please correct me if I spout nonsense here.
Chaitin and the Omega number is also an interesting follow up to Gödel/Turing and should be included but I guess he could be left out.
What is really fascinating about the Cantor-Gödel-Turing-Cohen chain imo is that they all had to invent new procedures to do their work. Diagonalization, Gödel numbering, the universal Turing machine, forcing. (and I guess lambda calculus if you include Church which I admittedly don't know much about). I think that would make for a much better overall theme than circumstances of death.
The one thing I really liked about that documentary was the shots of Gödel's scetchbook when he worked on normal stuff and when he worked on the continuum hypothesis. I tried to find that in image form via Google but no dice, maybe someone with better search skills can find it for me