On The Other Hand...

The death of the last Time Patrol Academy graduate might occur not at any point along the timeline guarded by the Patrol but in a divergent and deleted timeline. In the latter case, maybe the moment of that last graduate's final departure from the Danellian timeline counts as the conclusion of Patrol history? But then the Patrol agents who had deleted the divergent timeline would continue their careers.

Patrol agents from different periods communicate and cooperate so maybe there is no single last moment of their common history? There is a finite number of such moments even though what is the last moment for one is going to be an intermediate moment for another.

Also, even an agent or a Danellian who has read the entire history knows that, on any occasion when he travels into his past and returns to his present, he might enter an altered timeline contradicting the history that he remembers. This possibility exists as long as he is alive although it can be avoided just by staying in the present. Clearly, the Danellians want to be free to time travel without having to worry about returning to the wrong timeline.

This has to be the ultimate in "...curious possibilities of anachronism and of utter confusion..."