I'm facing the same issue as spidersilk in this old question - Is there a way to add a custom field to the Inbound SMS activity type?. Trying to keep the one CiviCRM instance I support as "out of the box" as possible, so hoping for an answer that doesn't require custom status-es or CiviRules.
Is there an "out of the box" way to search for activities with a blank "Assigned To?" With that, I could just list activities of type "Inbound SMS" that have yet to be assigned. I realize assignment isn't the end, but it'll suffice. I had high hopes for the Search Builder but, to my surprise, Assigned To is not one of the fields exposed for Activity.
CiviCRM 5.21.0 on Drupal 7.69.