While Star Trek: Discovery is not adhering strictly to 1966-1969 canon (for example, "Turnabout Intruder" establishes that there's either an institutional ban or a glass ceiling preventing women commanding starships!), the first time, in The Original Series, we see, using automated drone probes of any kind is "The Immunity Syndrome", stardate 4307.1, despite several situations where such a probe might have been useful. Even shuttlecraft are rarely used to extend the ship's sensor range (that same episode being a rare counter-example), unless the ship itself is needed elsewhere.
As such, we can safely assume that Shenzhou does not have robot or remote-control probes it can use. Further evidence for this comes from the fact that someone was going to have to manually pilot a worker-bee to deliver a torpedo. If Shenzhou had robot/remote-control probes, then a torpedo warhead could have been mounted in or on such a probe just as easily as it was beamed out to be planted on a corpse.
It's worth noting that this does not mean that Shenzhou has never been equipped with such a probe, or that they don't exist in this time period at all; only that none were available at the time they were needed. These stories pre-date matter replication on the level we see in TNG. It's possible that Shenzhou can carry such probes, but had used them all up and not been re-supplied.