The TSA permits "demonstration flights for marketing purposes" without jumping through the citizenship hoops but that is limited to "a flight for the purpose of demonstrating an aircraft's or aircraft simulator's capabilities or characteristics to a potential purchaser." See 49 CFR 1552.1(b). There is no exception in the regs for "intro rides" for potential trainees and I know of no exemption for such rides in the form of a TSA Chief Counsel letter. Other than giving them a sightseeing ride under 91.147, which means getting the FSDO letter and having the pilots in a random drug testing program and not calling it training or signing a logbook, it sounds like this operation is violating TSA rules unless they're complying with 49 CFR 1552.3(h) for US citizens and refusing noncitizens without the full TSA checks.
Not having an exemption for the "intro flight" is an idiotic policy, but it is the rule nonetheless.
Flight training, like everything else, has to be marketed. In aviation it's made more difficult not only by the expense, but also by the fact that not everyone who takes a "discovery flight" has even decided that they actually want to learn to fly. Some are just curious, some see a sign as they pass an airport and try it on a whim, some are interested but don't realize how expensive it is until after the flight, and so forth.
Some schools attempt to delay the TSA hoop-jumping by not calling the intro flight a "lesson" until after the student has decided to enroll, and not enrolling the student until the TSA requirements have been met. When I was shopping around for schools, the instructor at the first school actually read and had me sign a statement explaining that the "intro flight" was not a "lesson" and wouldn't count as such until and unless I enrolled at the school. Until then, it was just a flight whose purpose was to
help me determine whether I wanted to enroll. Until I actually enrolled (and completed the TSA requirements), nothing I did in an airplane would be considered instruction.
Another school I know of issues a laser-printed "First Flight" certificate (suitable for framing), instead of the traditional staple-bound temporary logbook, after the intro flight. The certificate is accompanied by a glossy digital picture of the instructor presenting it to the [not-quite-yet] student, both with suitably raised thumbs. The certificate also bears all of the required information for a log book entry --
except the instructor's CFI number and signature. So it's not technically a log book entry. It can be converted to one, however, if the student enrolls.
I was, however, asked if I was a citizen by all of the flight schools I considered; and the first school told me that I could take the intro flight, but would have to present my birth certificate before I could actually enroll. After the first school, however, I knew this and simply brought my birth certificate with me to the subsequent schools.
-Rich