After some trial and error, we sorted out the issue we were experiencing, as it appeared to be an issue with the user account, not with licensing or settings or anything like that.
In the Zoom Admin section, we went to User Management > Users and found the affected user(s). We edited the user settings to remove the Zoom Workplace license and set it to Unassigned. Then, under licenses and add-ons, we unchecked Zoom Phone Basic, and saved the changes (it appears that keeping this checked holds onto some Zoom Phone setting(s) that still prevented delegation). Then we went back into the same user's settings, re-added the Zoom Workplace license, and saved the changes.
Then, in Phone System Management > Users & Room, we found the affected user(s), activated them, assigned their extension and numbers, assigned their desk phone (if applicable), and we were able to delegate them to another user.