In my days it was a requirement to have enough fuel to get to destination, hold for 30 minutes, then fly to a diversion airfield and be able to hold there for 30 minutes there as well.
I have been flying since 1975 and 30 minutes holding at the alternate has never been a requirement in the UK.
If flights are arrving at JFK and declaring min fuel then they either have more fuel than they are indicating, so are not really in an emergency situation, or should have diverted earlier.
I wish life was that simple. And if you think it is that simple then you need to use the search function here on PPRuNe.
Sometimes one just gets a feeling pilots are declaring min fuel just so they can get to the gate earlier.
Fuel remaining goes in the Techlog. It is subject to inspection having declared an emergency. Once found out, and if caught "cheating" the FAA/CAA are going to have an uncomfortable conversation with the offending (ex) Captain.
It simply does not happen. Call me naive, but I don't believe it happens.