With the test data...
The "Status" at the top of the page will remain Unscheduled, even after a page reload:

The submission does not appear in the published table of contents.
It does not seem possible to resolve this, as there is no "Publish" button available in the workflow.
It turns out this was at least partially a permissions problem -- I was logged in as admin, rather than an (assigned) editor/manager.
However, I was confused by the "Unscheduled" state when the submission was scheduled against an issue (see screenshot above).
@NateWr, is this a PEBCAK or would a terminology change be helpful?
There should be a "Schedule for Publication" button that appears where the publish button would be. The article is not considered "scheduled" until that process has been complete. This ensures that the pre-publication checks can be run before a publication is ready to go, and a publication without a required pre-publication check is not published with an issue.
There is a fix for this permissions issue waiting to go in with these PRs: https://github.com/pkp/pkp-lib/issues/4874#issuecomment-567915219
(It's probably worth keeping this open to ensure the bug is fixed after those PRs are merged.)