Sure, I entirely agree on principle. But that’s my question, who are these people? According to Wikipedia, there are no such people:

Although Fuegians from Patagonia may have visited the Falkland Islands in prehistoric times, the islands were uninhabited when Europeans first explored them.

So I suppose I have my answer overall. The people living there are broadly the descendents of the French, Spanich and English who settled there in the late 18th century. The French ceded the settlement to Spain a year after they started, so even just scratching the surface of the situation it looks like it’s been a mess for 250 years.

But in any case, the people living there, now, clearly want to remain a UK territory.