Or they just ignore somebody shooting him for no reason also YAY!
Seriously sometimes I wonder if some people watched a different movie than I did.
They didn't know he had been shot. I'm wondering what movie you've been watching? They didn't know he had been shot until they had recovered him and brought him aboard the Enterprise
for inspection and repair.
And seeing Dougherty didn't seem to mind them poking around until they found the ship, your point is?
Besides why is everyone care about whether Picard shouldn't have gotten involved or not, everything worked out in the end.
The Son'a and Baku are trying to get along and the federation can just build a science station in orbit of the planet and study the rings so they can duplicate the radiation, and the Baku get to keep their planet. So everybody wins.