USA - All students must show "papers"

I happen to be a "resident alien" -i.e. green card holder rather than
a Citizen. I live here and have held a US pilot certificate for many
years both Glider and SEL.

There are a lot of people like myself who call this country home who
will have to jump through hoops just to get a BFR (I don't work for an
airline so doesn't appear to qualify as Category 4), or if we wanted
to work on a new rating, perhaps simply if I were to show up at a
glider operation for a couple of days and wanted a "check out" - that
would be a new training event at a new flight school - OK that'll be
$130, a repeat investigation and sorry sir you can't get a checkout
until we get TSA approval. Does that make any sense?, does that add
to flight safety or rather detract from it by discouraging additional
training?

As I interpret the ruling this is not a 1 time vetting of the person,
this is an approval process for each new type of "training" that the
"alien" seeks.