It seems questionable to propagate the parent span to jobs (callables and runnables) that repeat forever. This leads to traces that never close.
I have a use case where a first-access causes a scheduled job to be created 'scheduleAtFixedRate' or 'scheduleWithFixedDelay'. This is a repeated job, as such, it should not be associated with the parent span that initially caused it to be scheduled.
This behavior is in the TracedScheduledExecutorService class. I would suggest that inclusion of the these 'rules' be opt-in, not by default.
I'm curious what the use case is of associating a repeating job with a parent span, as I can't think of one immediately...