Featured Content: featured tag is hidden from editor when it isn't shown on front end

This appears to be a regression similar (or identical?) to #7996.

On themes with a Featured Content area/slider that's populated by a specified tag, checking the box labeled `Don’t display tag on front end.` causes the tag to be hidden from the post when editing in Gutenberg. I've just confirmed, the tag DOES appear directly in wp-admin if you pop open Quick Edit, but it's gone from the block editor.

Previously this was attributed to the tag also being removed from API calls which impacted remote editors. The symptoms appear the same here so I've reporting in the repo but if it's being caused by something else I'm happy to move those over to the Calypso repo.

**Steps to reproduce:**

1. Activate Twenty Twenty Four (or any other theme with a Featured Content area/slider that's populated by a tag)
2. Go to the Customizer and add a tag name for the Featured Content section.
3. Add a new post from the Calypso dashboard and link the featured tag to the post; publish the post.
4. Go back to the Featured Content section of the Customizer and toggle the “Do not display tag on front end.” setting, and
5. Save & Publish the changes.
6. Refresh the Edit Post screen of the post you just published in Calypso. Check the tag listing.
7. Repeat steps 3 & 4 by changing the “Do not display tag on front end.” setting.
8. Refresh and check the tag listing again. Notice that when the “Do not display tag” setting is disabled, the tag disappears from the Post Settings of the post.

Start by reproducing the tag visibility difference between the Gutenberg editor, Quick Edit in wp-admin, and the Calypso dashboard after toggling the Customizer’s “Do not display tag on front end” setting. Compare the relevant API calls and editor data, then verify that the featured tag stays available in Gutenberg while remaining hidden on the front end.

Written by the indexing model from the issue text.