It's always fun to analyze these sort of things to death. On that note...
1) As you can see, 1295 had already been an extra load on the TC because he had a number of issues going on starting up his "cold airplane", thus drawing focus by the TC (paying more attention to this guys plight than the others on the conveyor belt).
Someone on the ground at KFJK isn't perfect and had to be given an altered taxi route??? Say it ain't so!
While I don't doubt it's a nonzero amount of effort, I do doubt this "extra load" is really all that "extra" for a JFK cab controller. So he had to move the strip down a few slots in the bay, look around for a suitable holding spot in a movement area, and issue a few taxi instructions. If that was the beginning of the downfall of a JFK non-trainee... then may God help us all who fly into/through/out of KJFK.
2) I could not find any rollout instructions from the TC to Car 526... the last you hear is the TC giving 526 the landing clearance, which 526 reads back as 525.
Well yeah, he didn't give any extra instructions. But what's wrong with that?
Sounds like he heard Redwood's readback and then tried to get 1295 off the ground. Based in the gap in the transmissions, I'm guessing the distances led him to believe he had the time to fit the (already delayed) departure between the two arrivals without adding much more delay for anyone.
Seems reasonable to me to not solicit explicit hold short instructions from any aircraft exiting a parallel runway before issuing a takeoff clearance unless
the arrival was previously given ambiguous instructions or an instruction to cross the runway. Otherwise, on feeds like LA South Tower, you'd hear things like "(ACID), runway 25R, taxi full length, hold short runway 25R"
over, and over, and over again for every aircraft GC-2 transfers over.
Thus I believe the 526 pilot, expecting a rollout instruction, heard the instruction with a "six" as his and acknowledged making the same 525 vs 526 error he did with his landing clearance.
And this, IMHO, is strike #1 for the pilot who somehow made "RED WOOD FIF TEE SIX" sound like "CUH RIB BE UHN FI(F/V)E TUHWHEN TEE SIX" in his head.
The calls from the TC to 526 to "stop" as 1295 was on his takeoff roll were not heard because 526 had already switched to GND, which you will hear in the recording.
And this, IMHO, is strke #2 for the pilot who somehow decided it was good practice to leave the tower frequency before crossing an active runway with traffic
apparently lined up (but in reality, rolling) on it.
The TC's transmission was given in a fairly logical order:
1) Be alert for wake turbulence.
2) Cross runway 22R at Juliet.
3) Turn right onto Bravo.
4) Monitor 121.9.
I get the desire to expeditiously get on the GC frequency in order to receive further taxi instructions ASAP. But why would that lead you to do #4 first (or second), i.e. before reaching (let alone crossing and clearing) an active runway? Not only that, why would you immediately leave tower's frequency (e.g. as opposed to tuning ground in COM2 and switching active COMs over to it) before doing so?
Conclusion: The TC dropped the ball handling the arrival of 526, probably because he was so excited about 1295 finally getting out of his hair.
It's interesting how the exact same facts/stipulations can lead to two conclusions on opposite ends of the spectrum. Mine, for example, is that the TC did almost everything short of sending a follow-me car out to meet the Caribbean at the runway in order to ensure a competent operator was doing the taxiing. He didn't do that, but he did keep a scan on the runway and eventually spotted the pilot's error.