but the FAA Instructor who signed off that you were in fact ready for the Flight Test and who would have required the 3 hours to be completed prior to that test would have required that a TSA clearance had been completed.
That I agree with, for
him - not least because the last few hours, done typically within weeks of the checkride and
resulting in the FAA instructor's signoff, is rather obvious
But that is a world apart from
They wish to receive flight training from an FAA-certificated facility, provider, or instructor that could lead to an FAA rating whether in the U.S. or abroad.
which, taken literally, is completely unworkable and must have been written by somebody without a clue about the way flight training works, and without a clue about the very generous FAA rules on acceptability of training in other ICAO countries.
For starters, most UK instructors from a few years ago are not "around" because they got an airline job / moved away /etc. And even when you have traced him/her, verifying his
then qualifications/approvals is next to impossible. Funnily enough, my main PPL instructor vanished without trace (honestly) following some "events" at the school he ran...
Like a lot of flight time logging, most logbook entries have to be taken on trust otherwise the system would be unworkable. Take some ATPL (JAA or FAA) candidate; is somebody going to go through his logbooks and check all 1500hrs against airport movement logs?