Having operated out of JFK two days before the incident with the same runways in use I can see exactly how the Swiss cheese holes lined up. The incident actually started at the gate.
The JFK ATIS gives runway 04L and runway 31L for departure. You do not know which one you will get so you guess one and plan for that although you get the figures for the other runway too.This is the first Swiss cheese hole.
The datalink clearance comes through with a SID. However the SID, i.e. a Kennedy 5 does not specify which runway. At other airports around the world the SID name defines which runway you use. This is the second Swiss cheese hole.
So you brief for departure and push back from the gate still without knowing which runway you are departing from. This is not desirable.
When you come out of the ramp at TA you do not know if you are going to go clockwise or anteclockwise, it can be either, and so you are listening hard to whether it is Alpha or Bravo taxiway and where your holding point limit is.
Crucially the ground controller in the initial taxi clearance to AA106 did say runway 04L at the start of the transmission but this was not read back by the female pilot who just read back the taxi instructions. The taxi instructions they were given were relevant from their taxi position to both runway 31L and runway 04L. The next Swiss cheese hole lines up.
I think they believed they were taxiing to runway 31L for departure. They taxied as instructed on Bravo and held at Kilo. The next instruction they were expecting to receive was to cross the runway in front of them and continue to the hold for 31L. As a bit of confirmation bias they could see a plane at the hold for 31L.
As expected they then received clearance to "Cross runway...." and taxied forward but did not take the first right and went ahead believing they were cleared to cross that runway in their route to 31L. They did not realise they were cleared to cross runway 31L and not 04L. The final cheese hole lined up for the incident to happen.
I have sympathy for the crew and had they been given a departure runway in their original clearance and briefed the taxi route in advance before pushback I do not think this would have happened.