I think it is mainly for SSO, so if you log in to some other Google service, you are logged into YouTube automatically. Required because YouTube is not under google.com, it has its own domain, and for branding reasons Google wants to keep it that way.

Also, you can embed YouTube videos in some other Google services, and the SSO may help in making that feature work correctly. What if a teacher creates a private YouTube video and embeds it into Google Classroom?

It could be an issue with a overzealous blocker blocking not just YouTube video URLs but also a bunch of APIs and/or CDNs known to be used by YouTube, which also happen to be used by several other Google services.

In Google's defense, they have no reason to set up a different API or CDN for each service they own and operate. That would be a maintenance nightmare.