First of all, there are trillions of people in the Alpha Quadrant. They only have to convince (or rather entice to look at their evidence) a tiny sliver of the population to get the ball rolling.
While a tiny sliver of trillions of people is a huge number, the remainder who aren't convinced is an even huger number. And on the other side you have the Zhat Vash, which is a small, secretive cabal that has extremely stringent requirements for induction into their ranks (don't go mad from the Admonition), which you yourself point out. You can't convince trillions of people with a few handfuls of people.
Are you suggesting that most scientist would actually accelerate their efforts to make synths if shown their evidence? That isn't at all obvious nor is it what we see happen in the show. The only person they share evidence with actually tried to murder a man to prevent it from happening.
As I said below in my post, the rock-hard evidence the entire Zhat Vash rests upon is a telepathic vision. If a person was just told what the vision is about or shown a video without context, they're fully capable of calling bullshit because they're emotionally and contextually disconnected from it. Jurati was convinced because Oh mind-melded with her, which has been consistently shown on Star Trek to transfer emotions and thoughts along the visions. Jurati received a much deeper understanding of the Admonition than someone merely hearing about it would.
Fist of all, let me say how convenient it is for the plot that the simple facts of the story, conceived by mortal writers in the real world, will destroy the minds of anyone who sees it first hand as if they were staring down Cthulhu himself.
We literally saw Romulans clawing their eyes out and shooting themselves in the head after seeing the vision, so yes, they were basically staring down Cthulhu himself.
You're also assuming that no one would believe it unless they witnessed it first hand themselves
If someone merely told me "this is what would happen if [insert group here] were allowed to exist" and then showed me a video of palents being blown up, I'd laugh in their face, call them a racist asshole and walk away.
There are also ways of mitigating risk, such as showing stronger-willed people who are more likely to survive (such as Vulcans), disarming people before having them touch the device so they can't shoot themselves right afterwards, having people standing by to restrain and sedate people right afterwards.
Which the Zhat Vash themselves were shown not bothering with. It was more like an initiation ritual. If you're strong enough not to go insane, you're Zhat Vash. If not, you weren't worthy enough to begin with. Quite possibly the only people who are ever shown the Admonition are the inductees and, in more recent times, whoever Oh wants as an operative.
There's also the fact that Jurati was convinced by a completely survivable mind meld.
Which was relayed through a Vulcan who has already seen and survived it. When someone tells you a story, you don't experience it as it happened, it's filtered through the person who tells it. And it's obvious that Oh is in a unique position because she's able to mind-meld with people to show them the Admonition without making them insane.
The only person we see them share evidence with was a human who found it convincing enough to try and murder a man and also influenced others to discover the octanary system they were trying to hide for some reason. What you insist is ridiculous and implausible is exactly what we see unfolding in the show.
I was talking about people who were
told about the visions or shown
footage of it. You're talking about someone who was subject to a
mind-meld by the one single half-Romulan member of the Zhat Vash.
