Someone posted a possible solution here:
It’s in Japanese, but a quick translation:
Reason:
This is because Obsidian has a third-party cookie for the YouTube domain, which causes the error.
Workaround:
Open Chrome DevTools in Obsidian and delete your cookies.
- Launch Obsidian and open the (notebook) vault where the error appears.
- Open Chrome DevTools. Open the Application tab, expand Cookies, and right-click any YouTube cookies to select Clear.
Once the Cookies are cleared, you’re good to go. Close the note and reopen it, and the video thumbnail should appear.
I also suspect it may, in part, have to do with region/country/geo-locking for embeds, but that’s just a guess.