what a story
your question is perfectly reasonable. Air Traffic Control would have a huge book of every conceivable operator of airplanes and they would have to know which particular plane and crew were operating on a particular segment.
At my airline of over 5000 pilots, some may have not completed CAT II or CAT III training. should atc have a book that says, Everyone at Airline "X" can do CAT II except FRED?
It is just impractical for ATC to do the policing of such things. Mind you I think there could be a better way. Some sort of information filed in the flight plan, and presented on the radar display...but they don't.
AND to make things even more difficult, some forms of regulation do not have the OFTEN MENTIONED approach ban. Indeed there are airports out there that might not even have weather information or local ATC service at all.
Good for you for asking.
Wouldn't it be odd if someone in the training department at whatever the airline really is actually told these poor pilots they were CAT II? We will never know.
I remember flying the Metroliner back in 86. We were poorly treated and the dumb girls who ran the ticket counter told us that our little airline could still exist without pilots...because they would just sell tickets on other airlines.
that's the way things are...little airlines are often crap...not always, but there are shortcuts!