The catalog would be written to tape regularly: this is what would gets e-mailed out. But it wouldn't necessarily be written to every tape.
Remember that the catalog changes every day: you'd have Version 3142 of the catalog at the beginning of Monday, but then you'd back a bunch of clients, so that catalog would now be out-of-date, so Version 3143 would have to be written out for disaster recovery purposes (and you'd get an e-mail telling you about the tape labels and offsets for it).
In a DR situation you'd go through your e-mails and restore the catalog listed in the most recent e-mail.
I'm with the OP - depending 100% on a file that's not on the tape to restore the tape is bonkers. It's fine as optimization, but there should have always been a way to restore from the tape alone in an emergency.
One potential trade-off is being able to write a continuous datastream relatively unencumbered vs having to insert data to delineate files, which is going to be time consuming for some times of files.
Right but is that actually possible? From what people are saying it sounds like it isn't, but you rightly assumed that it is because anything else would be incredibly dumb.