People who flush their passports on board and then arrive at passport control "undocumented," claiming to be from a country for which the USA offers asylum.

Happens more than you might think.

I have gone through federal SIDA training and can assure you that TSA staff does not have authorization to enter any airport jetway unescorted, much less one within a sterile international arrivals area. The OP must be referring to either CBP or ICE, both of which are federal law enforcement agencies charged with enforcing immigration/customs laws. On several occasions arriving from Europe, I have had to show my passport to ICE and/or CBP officers right at the door of the aircraft.

TSA is not a law enforcement agency, and has very few sworn LEOs on its payroll. TSA would not be conducting any checks within a sterile international arrivals area, because TSOs are not authorized to work in that area.