You do have the advantage of Spanish as a native tongue here - does he really declare a fuel emergency? To formally declare an emergency, the "Mayday mayday mayday" call would be necessary.
From the
subtitled ATC recordings I have been able to find, I have gathered that he requested priority because of a "fuel problem" (in that video at about 0:20). This is not a declaration of emergency and would be understood by controllers (at least around here) as a heads-up that they are approaching minimum fuel. At their flight planned destination, this would be alternate fuel (to get to their nominate alternate) plus final reserve (usually 30 minutes) left in the tanks.
Had they declared emergency earlier as they should have, possibly stating how much endurance they have left, it would have been a trigger for the controller (and with regards to her, I agree with the previous posters that she sounds like a true professional and hope that she will get the support she deserves) to get the other aircraft out of the way to get the LAMIA in ASAP. The outcome may very well have been different in that case. But as it sounds, they kept that vital information from the controller right until the point where the tanks were empty and the engines were flaming out.