So I have more things in mind.

Basically PersistentTaskGroup resemble TaskGroup in that:
 - It has the same "create_task()" method.
 - It has an explicit "cancel()" or "shutdown()" method.
 - Exiting of the context manager means that all tasks of it have either completed or cancelled.

TaskGroup is intended to be used for a short-lived set of tasks, while PersistentTaskGroup is intended for a long-running set of tasks though individual tasks may be short-lived.  Thus, adding globally accessible monitoring facility for plain TaskGroup would not be that useful.  In contrast, it is super-useful to have a monitoring feature in PersistentTaskGroup!

In aiomonitor, we can enumerate the currently running asyncio tasks by reading asyncio.Task.all_tasks().  This has saved my life several times when debugging real-world server applications.  I think we can go further by having asyncio.PersistentTaskGroup.all_task_groups() which works in the same way.  If we make different modules and libraries to use different persistent task groups, then we could keep track of their task statistics separately.