The Falklanders are totally dependent on the UK government. There's no economy that is self-sustaining by todays standards. Forget the maybe-oil-decoys.
Immigration rules to the Falklands are probably one of the most unwelcome for any non UK immigrant in existence.
I am not for or against the Argentine claims, but the UK's protection of the peoples holy rights on self determination seems cherry picked (
The United Nations and Decolonization) and selective (
BBC News - Mass protests as Hong Kong marks 15 years under China).
Asking these people if they want to get disconnected from the UK is like asking a drowning person if he wants to sell his life jacket. And everybody knows.
I would be pragmatic and offer all the Falklands a better house/ job somewhere in the beautiful roomy UK and make a deal for those that want to stay. The youngster move out anyway.