When claims are made by a politician, I wait for other sources; independent sources with nothing to gain by such statements or clout to buy from voters.
I looked at the Schumer claim about flight students the same way. I've now seen how two large schools with a large number of international flight students handle meeting the TSA requirements. It's done quite well. The funny part is, the TSA follows the paperwork with a close eye. The students are legallly enrolled at those flight schools. The TSA's eye on the actual student is another story.
I'm no fan of government agencies. But, I'll take the word of inspectors and those working the "front lines" on such issues over politicians or political appointees and immediate subordinates; any day of the week and twice on Sunday!
