This reduces the number of retries that might spam APIs.

There is some complexity here which is left un-managed -- for
instance, maybe_restart_mirroring_script does a number of restart
attempts, and then fails, but will be retried every 15s by the
surrounding `process_loop`.  Previously, it would merely have looped
forever inside maybe_restart_mirroring_script.

Three loops are intentionally left as infinite `while True` loops,
that merely cap their backoff at the default 90s.  Their callers do
not expect, or have any way to handle more gracefully, a failure of
the expected-infinite-loop in `process_loop` or `zulip_to_zephyr`.
They maintain their previous behavior of retrying forever, albeit more
slowly.