Dougherty's rationale for forcing the Ba'ku to relocate rather than share is a fairly weak one, but we don't hear the full set of evidence he believes in. The Son'a were lying to him about many things, and might have convinced him that sharing would be really bad not just for the suffering Son'a, but for the Ba'ku themselves as well.
We can rather safely assume that the UFP Council was told the same lies that Dougherty heard.
Picard never is in on the loop where the fate of the Ba'ku or their Fountain of Youth is decided; he comes into the picture way too late for that. Dougherty's not really obligated to explain himself or the Council to Picard, and he's pissed off with the Captain's antics already, so it's pretty natural that he doesn't expand or explicate. Which serves the thematic structure of the movie very well.
In fact, given that it was an entire planet and they represented only a small village, there's a good chance they wouldn't even know if others settled it, say, on another continent.
They didn't realize they were under surveillance by holo-invisible aliens. OTOH, they quickly diagnosed what was wrong with Data, even when LaForge always requires lots of equipment to diagnose his long-time android friend. So we are left pretty much in the dark about how much of their original technology the Ba'ku retain and keep in use, and whether they could use it to spot intruders.
We can't even tell whether they can spot starships in orbit, because the only ship in the region before the exposure of the alien observation plot, the Son'a command ship, might have been keeping her distance. Even advanced technologies would have trouble spotting distant ships in that soup.
Speaking of the soup that was the Briar Patch, the Ba'ku apparently arrived around the 2070s. As of the 2150s, the area was apparently Klingon property, as per ENT "The Augments". It seems doubtful the Ba'ku settlement of a Klingon world (no matter how unexplored by the Empire) would have been legal, so the legal intrigue centers on what happened when the UFP annexed this space. Would any inhabitants at the moment of annexation have been declared native owners of the land, regardless of their history?
An ending where others come to establish colonies or make use of the particles would have gone a long way to make the Baku more sympathetic.
Very much so. And it could have been softened by some technobabble about how the rejuvenation doesn't really take and the "patients" or "spa customers" only get temporary or partial relief unless they agree to live the rest of their lives in the Briar Patch - still much better than nothing, but no longer something star empires would slaughter each other for.
The cut ending where Quark arrives with the intent of establishing the first spa would not have addressed these points satisfactorily, though.