A better way to think about this is to consider the case of Saint Pierre et Miquelon, which are two islands off the coast of Canada -- literally 19 km away, and so much closer than the Falklands are to Argentina. They are French territory, inhabited by French citizens. And (aside from occasional disagreements about who gets to explore for undersea resources in the vicinity of the islands) Canada really doesn't seem to have a problem with this, because (in this area, at least) Canada is a more mature nation than Argentina.
How does that work? Is it some kind of algorithm? Does the USA have the right to own Canada?
https://www.reddit.com/r/MapPorn/comments/coxtsv/countries_i...
Of course, there's some hand waving here - micronations are dropped, Mercator projection, etc
And where does it end? New Zealand belongs to Australia? Hawaii belongs to Mexico?
In short "proximity" is not how island ownership works.