Use the new task monitor for long-lived background jobs

[bnjbvr]

#6075 adds a mechanism to watch background jobs spawned with tokio::spawn that have failed (either with a panic, or with a Result::Err). The same PR introduced its usage in the event cache, but we should expand it to more places in the SDK, i.e. for every background task.

This is rather mechanical, and requires a bit of thought for each case. In particular, if a job is spawned in the background but awaited immediately, it should not use this mechanism; this is really for jobs that ought to live ~forever until the program dies (or aborted in rare cases).