To answer the initial question of if they just got lucky with the timing, I'd say we don't really know; but we can speculate a bit (or a lot apparently, based on the size of this answer).
It seems highly improbable that the secret societies on Roshar, such as the Sons of Honor, Ghostbloods, Skybreakers, and The Diagram, happened upon their information by accident or coincidence. Just based on the goals and methods of the Sons of Honor, it seems pretty clear that they are working off of very limited information, and are perhaps being manipulated in their ignorance.
For example, in the opening scene in The Way of Kings, there is a lot of detail put into the description of several attendees at the party. If you pay attention to what is being said, and compare it with information in the later books; it becomes clear that several people in attendance were Heralds. This basically makes the entire plan a waste, and shows a good example of how misguided they were in their understanding.
It also seems as though if they would have accomplished their goal, they would have only really ended up with a False Desolation. This is because what they were doing seemed to heavily involve the black sphere, which from OB we can assume contains an unmade. This means that it was either Ba-Ado-Mishram, and they were creating a false desolation, or it was some other unmade, which shouldn't produce any desolation from what we have seen.
We also know Gavilar was getting visions from the Stormfather, which confuses it even more. This could imply that he was getting the knowledge for Sons of Honor from the Stormfather; but then raises the question of why he was given so many incorrect premises to work off of.
From my interpretation, they are essentially the opposite of the Skybreakers. The Skybreakers are trying to prevent a desolation by killing off Radiants. Sons of Honor were trying to create a desolation by bringing back the Voidbringers. Neither were completely noble goals, and neither seem logically consistent with what was holding back the desolation.
These organizations are being led by people who seem to have, or should have good information about what is going on. Gavilar had access to the Stormfather and his visions, and Nale is a Hearld and should know how the oathpact works. That leaves us the reader to guess if there are more mechanics to this that we don't understand, or if it's just a result of Nale's insanity and Gavilar's lack of information.
It seems very suspicious that these events all took place around the time of the everstorm/desolation, but the evidence seems to suggest that it would have taken place regardless, as Taln was the deciding factor.
It seems like there were forces acting to put everything in place for something to occur (and if you're familiar with Mistborn, you're probably aware of how subtle, manipulative, and good at planning ahead Shards can be). I don't think we're supposed to understand why all these events are happening, but from my understanding, the attempt to bring back the Heralds was not directly relevant to the start of the desolation.
It all seems very suspicious, and I believe we should suspect more than mere coincidence; but at the same time, it wasn't really possible for people on Roshar to impact Taln's situation on Braize. I wouldn't call it a coincidence that this begun shortly before the desolation; but I would also say that they were not the cause of it. I would guess that they were put into place for another purpose that has yet to be revealed.
I apologize for how long and potentially rambling this may seem. There's a lot going on, and I probably missed a million different threads; but hopefully it sheds a bit of light on why it seems to be a coincidence so far in the narrative; but likely isn't completely.