Site icon image in the Site editor doesn't update on the fly

[afercia]

Description

In the Site editor, when setting the Site logo block image to be used also as a Site Icon, the image is supposed to replace the WP logo used in the 'Open Navigation' control at the top left of the editor UI.
I would expect the image to update on the fly when it is set, replaced, or removed. It doesn't update. As such, the editor UI may show two different images for the Site logo and Site icon which feels confusing for users. Example screenshot:

I'm not even sure using the Site Icon image in the editor UI is a good idea in the first palce, for the considerations made on #57813. However, if keeping using it is the desired behavior, it should work as expected.

Step-by-step reproduction instructions

• Go to the Site Editor > Edit mode.
• Select the Site Logo block in the header or add one.
• Add a Site logo image.
• Check 'Use as site icon' in the settings panel.
• Save.
• Observe the image shown in the 'Open Navigation' (which is supposed to be the Site Icon) doesn't update.
• Refresh the page.
• Observe the image shown in the 'Open Navigation' is now the one you set as Site icon.
• Select the Site Logo block in the header.
• Replace the Site Logo image with another one.
• Save.
• Observe that while the Site Logo block image does update, the Site Icon image does not.
• Refresh the page.
• Observe the Site Icon image now the new one.
• Select the Site Logo block in the header.
• Uncheck 'Use as site icon'.
• Save.
• Observe the Site Icon image doesn't update. Expected: the image to revert to the default WP logo.
• Check again 'Use as site icon'.
• Save.
• Select the Site Logo block in the header.
• Open the 'Media' dropdown in the Settings panel and click 'Reset'.
• Save.
• Observe that while the Site Logo block image gets removed, the Site icon stays.
• Refresh the page.
• Observe the Site Icon image is now back to the default WP logo.

Screenshots, screen recording, code snippet

No response

Environment info

No response

Please confirm that you have searched existing issues in the repo.

Yes

Please confirm that you have tested with all plugins deactivated except Gutenberg.

Yes