If you are going on a 1000nm leg as in your example, you have plenty of time to use your resources and come up with a working solution before you have to declare min fuel. There are plenty of usable airports along the way that you could possibly use to get more juice if needed. Look at it this way....You have lots of time and options. Lets say 10 options to start with. The longer you do nothing, your options will begin to dissappear. The less options you have available, the more likely a min fuel situation will be declared. Pretty soon you'll be left with only 1 option.
There is a reason we check our fuel status regularly and compare our FOB to our calculated burns on our release. Any time our FOB is less than planned at a specific waypoint or if our EFOA goes below our planned FOA, it requires an acars to dispatch letting them know whats going on. If you catch it early, a min fuel situation can most likely be avoided. ATC delays ect....are a whole new ballgame and will be dealt with slightly different.