It was a bluff because either you're in a fuel emergency, or you aren't. He was trying to scare the ATC worker hoping they'd panic and slot him in to "avoid" the emergency. But that's not how any of it works. That's why it was a bluff.
The ATC being 4 minutes "late" for a callback during the busiest landing time does not mean the plane was "forgotten". 34 minutes is not very long to spend on hold, planes carry hours worth of reserve fuel. And estimates are just estimates, not a promise (like the article framed it as).
Honestly this is akin to someone showing up to the most popular restaurant in New York 2 hours late for their reservation, asking for a VIP room, then getting huffy when they don't have a VIP room within 30 minutes and then being angry they were offered a seat at the restaurant next door. (apparently the pilot, on the PA to passengers, was talking about taking legal action against ATC on the way to Oakland... ridiculous...)