Hell they could probably replicate the radiation particles if given a couple of months to study it maybe a few years tops. I mean these are the people who once invented a device makes freaking planets from nebulas and it wasn't even designed to do that (which is probably the real reason said planet exploded). Not to mention they already figured out ho to make the transporter de-age and re-age people under certain circumstances so making a fountain of youth should be a freaking cakewalk.
More importantly, extracting the particles ignores the one giant elephant in the room: it
is instantly turned into a finite resource.
Who gets it?
Loaded language like "cure for cancer" completely circumvents the fact it will only be a cure for a select number of people. Everyone else will have to stick to chemo.
How does anyone decide who has a right to it without creating the timeless segregation of have and have-nots--the fundamental dilemma the Federation was created to eradicate?
The only way to prevent this from happing is to, as you say, find a way to replicate it. So why not just do that on the planet and live and let live?
Ultimately, finding a way to extract the medical application of the particles would only be a matter of time, of which, incidentally, any researcher left to his devices on the planet would have an unlimited supply.