The US base has been established since 1971, but the rest of the Chagos are unpopulated. Mauritius maintains a claim on the Chagos based on the fact that in colonial times the barely inhabited archipelago was administered from there although the islands are geographically closer to India. India put in claims for the Maldives and Andaman Islands in 1947--the Maldive claim was ignored (the place was a protectorate) and India eventually got the Andamans in 1950. India ha a very big naval presence in the Indian Ocean (where else?) an regularly shows the flag at the Indian Ocean islands. Practically they would not take on the US, but sovereignty doesn't mean a lease could not remain--it was due to expire in 2016 anyway.
If the US can have a legal base in Guantanamo, Cuba for which it has been paying rent for the past 60 years or so since the Cuban Revolution, why couldn’t the US continue occupying its base on Diego Riveraregardless of who owns it?
There was wider diplomatic/political reason to defend the Falklands. If a major power,, however reduced in circumstances allowed a tin-pot dictatorship to grab its territory by force without reaction, what was (in 1983) to stop China wasting time on negotiation and just have the PLA walk into Hong Kong the next day (while the banks still had cash and gold in their vaults) or Spain to do the same in Gibraltar. Maybe India fancied Diego Garcia , Turkey would love to re-claim the Greek Adriatic islands off of its coast and maybe the Southern half of Cyprus too. China has enough claims along its borders and in the South China Sea to keep half of Asia awake at nights.
They don't. Stubborness and not wanting to look weak. The core fact of the UK claim is no one lived there when the UK got there and the core claim of the Argentine claim is that the "people" wanting to be in the UK is kind of irrelevant cause there's almost no people there and it's not a self determination issue.
There are gas and oil deposits (not sure if they are exploiting them or any of the technicalities). The Falklands actually would in the very long run become an issue due to the Antarctic Treaty, notice Argentina and Britain (and Chile) all have overlapping claims which would mean a clash the second the treaty is finished, whenever that happens.
For what it's worth, the argument 'they are close to us', is a bunch of garbage, France still has a literal colony in South America (French Guyana home of the European space station launch pads apparently) and I've never heard anybody have a problem with that, so I've never understood the generalized south american emotional support for Argentina's claim.