Is that really what they did in that episode? For some reason I remember it as just a small group that got killed. Yeah, allowing a whole civ. to be extinguished sounds dumb, especially since it won't provide any "evolutionary" or cultural value to either those people or others in the galaxy.
Yep. Worf's brother was assigned to observe and study this civilization, and then a random catastrophe threatened to wipe them all out. Worf's brother yelled "run to the caves" (buying them some time), sent a distress signal, and waited for Federation help to arrive. Worf's brother might have only saved one village out of an entire planet.
Then Picard tells him that he refuses to help these people, because death is better than a contaminated culture, and Worf's brother was wrong to yell "run to the caves", because even that warning contaminated them.
The Picard-ordered solution is, leave those people in the cave to die, while you stand on your bridge, salute, and say "This hurts me more than it does you, and I respect your unwilling sacrifice."
Then Worf's brother beams the survivors up to the holodeck, with a plan to relocate them to a safe planet without disturbing them, and Picard is all "Damn you for making this my problem." Picard was
completely willing to let them all die (through deliberate inaction), but he's not willing to pull the trigger himself.