Editor: Publishing post with featured tag doesn't add it to featured content section

Expected behavior

If my site's theme has a featured content section, and I publish a post with the featured tag, that post should be added to the featured section on my site.

Actual behavior

The post is published with the correct tag from the app, but it isn't added to the site's featured section until I re-publish it from the web. This may be specific to the Textbook theme.

Steps to reproduce the behavior

1. Enable the Textbook theme on a test site.
2. In the site's customizer, set a featured content tag.
3. Publish a new post with the featured content tag.
4. Check the front end of the site. Result: The post is published on the blog with the correct tag, but it isn't featured.
5. On the web, open the post in the editor and update it.
6. Check the front end of the site again. Result: The post appears in the featured section.

Tested on moto e5 play, Android 8.1, WPAndroid alpha-142

Internal ref: 1656900-zen

Start by reproducing the publishing flow described in the issue with the Textbook theme, a configured featured-content tag, and a newly published post. Compare the app's initial publish result with the later web update, then trace the Android publishing entry point to determine where the featured-tag state diverges. Done means a newly published tagged post appears in the site's featured section without a web re-publish.