Cover block: Videos can't be played in private sites

[fluiddot]

Describe the bug

Videos uploaded to private sites are not displayed when inserting them into the Cover block. Note that if the site is public, videos are played as expected.

To Reproduce
Go to a Simple site in WPCOM.
Go to the Site Settings screen.
Change Privacy to Private.
Open a post/page.
Add a Cover block.
Upload a video from the local device or record one.
Once the upload finishes, observe that the video doesn't load or play.

Expected behavior
All videos inserted on the Cover block should be displayed.

Screenshots

Public site/video	Private site/video
ios-cover-block-public-video.MP4	ios-cover-block-private-video.MP4

NOTE: In the private site/video case and concerning the video not being displayed in the first attempts to preview the post, it's likely that it's produced due to the video not being processed at the time the preview loads.

Smartphone (please complete the following information):

• Device: iPhone 11
• OS: iOS 17.0.2
• Version 24.2

Additional context
This issue is similar to #5497 and WordPress/gutenberg#59374.