While I basically agree with the commenets made about the speed of delivery in of some US controllers in busy airports and I would add even on some en route sectors, I do not think it is the main issue here. We could also debate here on how the ICAO ELPAC ( the language proficeinecy tests) were embraced by the US and the FAA when they became mandatory in 2011 but, it is another story , but the fact here that I believe this had very little to do in this incident . As correctly pointed out by BFSGrad , the quality of the internet audio recording most likely comig from a hand held cheap scanner in aparking lot, and would not the same quality as what you hear on the cockpit VHF,,. In addition the AA crew did not question the taxi clearance and everybody involved was a native english speaking American.
From what I read so far, the new briefing and checks to be performed while taxing, perhaps having to deal with another issue or having a discussion during that time made them misinterpret the crossing part of the clearance.. Unfortunately the CVR will not give us this info and most importantly the timig of things, so the incident report will have to rely on memory of the involved persons days afterwards .not really ideal to draw conclusions.