I am not aware of finder's keepers as being the determinative law. However, I don't believe that the British were the first to set up a settlement there. It looks like the French were the first.
Multiple settlements were setup. In different locations. Different countries.
At the time, there was no Argentina.
In 1829, Argentina requested permission from the British to try to setup a settlement.
The first two failed and the third was supported by the British.
When the government on the islands started to exceed their bounds; the British stepped in.
Geographically, they may be closer to Argentina.
In that case; why do we (US) have the Marshals, Hawaii, Puerto Rico, Guam, etc.
We were not the first, nor the closest country to any of those land masses.
There are islands in the Bering Sea that are closer to Russia than Alaska, yet they belong to Alaska