So, you've got an authentication layer, which can pass a token to the browser or physical hardware client access device. This should be very lightweight, no? Even with millions of users, this database shouldn't be enormous.
And then you've got your encoded libraries. This should be data being spit off platers on CDN networks. CDNs aren't hard anymore. Cloudfront is cheap, or you can go with Akamai, Limelight, Level3, etc. Properly built, you should never NOT be able to serve from somewhere in the CDN, even if the serving location isn't optimal.
Yes, this is HEAVILY simplified. I've left out recommendations, their encoding process/infrastructure, etc. This is not atomic data that is being replicated in realtime; these are videos that are lazily encoded, stored, and then served on demand.
It should not be this hard for this use case.