Wrong, they totally do. They run literally fucking everything in New York and New Jersey, from airports to actual trains to buses to, yes, subways like the PATH trains. If you aren't driving in your own car, you're almost guaranteed to be on something Port Authority runs. Even if you are driving in your own car, they probably have some power over you. Are you in a tunnel? Port Authority runs that shit. It is a ludicrously powerful and all-encompassing government authority over every single aspect of going from point A to point B.
(You may be right about general subway shit. . .there's a separate authority for that, the MTA. They're equally evil.)
The reason why the airport is more likely is not just because of the involvement of the MTA in other forms of pub trans, but also simply due to legislation. When she was presumably leaving Texas, they are likely to be much more lax about this sort of thing (assuming she brought it with her and then was leaving NYC with it). From what I know, when you touch down in an airport in the USA after a domestic flight, there are very few hurdles preventing you from leaving the airport as the assumption is that all of your stuff was checked on the way in. However, on the way out of NYC and getting on a flight, you are much more likely to be caught having something you are not supposed to than you are in Texas, due to the differences in legislation and culture surrounding weapons, to my understanding. EDIT: Before it is brought up, I understand that flight safety is not state-specific per se, but I propose that flight safety management is more likely to be influenced by local customs and procedures.
Another reason why I find the airport to be more likely than the subway or any other form of pub trans is due to simple logistics. While there are few hurdles preventing you from leaving an airport with a weapon after a domestic flight, there are even fewer preventing you from getting on a bus, train, or subway with said weapon. There is practically no security whatsoever, to my knowledge. No baggage check, no metal detectors, no air marshals or stewardesses, etc. She would have had to pull it out in plain sight of either an employee or an officer of the law, and even though we know she is fucking stupid, I doubt she is THAT fucking stupid.
This is speculation of course, but given that her booking was mid-day on a Monday, it is likely that this happened before a flight out of NYC. I imagine she spent the weekend and possibly some time prior in NYC, and then was planning on flying out in the morning (thus getting booked at midday - which would be some time after getting detained) and returning in the afternoon/evening to TTU, the possibility of which is further bolstered by the timezone difference. If we assume she did not have any Monday classes or exams, this would fit well with her schedule. She was apparently taking or preparing for exams in late July/early August IIRC from the logs, and because of the nature of summer courses, it is more likely that this may not have impacted her schedule. However, she got booked, and may have missed her flight as a result - I am struggling to confirm flight times to Dallas/Fort Worth (nearest major airport) from LaGuardia or JFK on that specific date (the 26th), but they seem to be relatively regular and consistent. Given her mother's connections, she may have even been able to fix this itinerary after the arrest, or have her transfer to a flight to Lubbock, TX area (there is a minor airfield there). Again, this is all speculation, but I believe the first two paragraphs are logically sound.