Arguing the "medical benefits" and who ultimately gets them is really sort of pointless They really aren't what this whole thing is about. Whoever ends up with them pretty much wins ... everything. A secret like that doesn't stay a secret for very long--not like it would if it were left on a backwater planet in a remote, inhospitable part of space.
The Son'a wouldn't have to tell the Founders. They'd find out quickly enough on their own, as would the Klingons and the Romulans and the Cardies, and the Breen, and the Tzenkethi, and whoever else is running around out there.
This thing is so big that it's going to be contained to the Alpha either. Queen Steelpants, in her endless pursuit of perfection, will assemble her legion space zombies immediately. The Vidiians? Of course they're going to explore every dubious mean necessary to find a way to the Alpha quadrant.
Hell, let's throw, Vulcans, Andorians, Rigelians, etc. into the mix. Because, let's be honest, when we say "Federation" in this context, we really mean "humans."
And it isn't just about ensuring control over the particles. There's also the matter of the perceived threat they represent. Creating a Khan army is the best case scenario. At worst, we're talking bout creating an entire sub-race of demigods to whom the entire Milky Way (and probably parts of Andromeda) instantly becomes one big sandbox.
Even The Q have to perceive this as a threat. After all, what is omnipotence if not unlimited evolutionary potential over infinite time? These particles solve the time problem, and they take a huge slice (like a few hundred millennia's worth) out of the potential pie.
In the end, we're left with one massive slippery slope that dumps strait into a bottomless quagmire and potentially the utter annihilation of Earth. Uber panaceas would seem pretty pointless then.
To put things into perspective, if some universe-shattering resource were discovered on Earth tomorrow, all economic and military alliances would be completely thrown out the window. Mutually assured nuclear destruction would be rendered meaningless. Basically, we'd be living in one massively fucked-up world. (Hey, I just had a great idea for a novel!)
Anyone who doesn't understand that this scenario is not only plausible, but probable, is being naive and short-sighted. It can happen on modern day Earth, or it can happen in a mythical interstellar future.
Really, the best course of action for all parties would have been to say "Neat!" and turn the fuck around and go home. Contain the Son'a. Want to put the "needs of the many" to good use? Shoot Ru'afo in the face. Gallatin and friends can go home, and the non-Ba'ku Son'a committed enough crimes to have them locked-up in a far-off place forever. Destroy any and all evidence the place ever existed, and hope for the best.