Former FA here so I will speak to cabin crew issues only.
I flew for an major US airline where flight attendants have full authority to initiate an evacuation. All of the comments in the thread about pilot checklists and ECAM messages were irrelevant in our training. If I saw significant fire that I thought threatened the lives of the passengers, my immediate action item was to hit the evac button and begin the evacuation. In this case, though, it sounds like the captain initially commanded the crew to remain seated - most likely due to what folks mention above such as the danger of jet blast, the danger of being run over by ground equipment, etc. But once they had significant fire on the starboard side and passengers starting to panic, as I flight attendant I would have initiated the evacuation, period. Shrapnel could have punctured fuel tanks, and there could have been a fuel source for the fire under the wing and fuselage that could have led to rapid intensification.
My first thought, though, is on these densely configured airplanes how little access to outside information the cabin crew has. The tiny windows in the exit doors may not have provided sight lines to that fire. And given the panic in the cabin, it may not have been easy to get egress to assess the fire. My hunch is that it was the passengers - and not the pilots or flight attendants - that had the clearest knowledge about the most severe phase of that fire.
The PA announcement about firetrucks being on the way is baffling to me, though. If you have a fuel fire with long orange flames and thick black smoke, my job as I understood it was to evacuate the airplane immediately to buy precious seconds to spare people's lives in the case of rapid fire intensification.
Assuming you were TWA, there are couple of significant differences from EVACs of then (I'm somewhat familiar with TWA procedures at a second hand) and what the current best practices say.
Especially in the A320, jet blast represents a danger to overwing and rear doors/slides, the jet intake represents a danger to the L1/R1 doors, and the way the doors open represent a danger to the F/As if the aircraft is pressurized (there have been multiple incidents where serious injuries have occurred to F/As and gate agents when opening pressurized Airbus doors.) Next time you're boarding an Airbus notice how the doors open out and forward and the pressurized light viewable via the door window.
The pilots have pretty good view aft in a 319/320, a little less so on the 321. Ultimately, the biggest dangers during the evac is not having the engines stopped, and responding equipment target fixated on the fire and not the passengers milling around, filming, helping, finding loved ones, etc.