YouTube is estimated to have 1 exabyte [1] of data. Petabyte level storage is not unheard of [2], and a gateway server with 5PB storage would cover ~0.5% of all YouTube videos, which should be sufficient to serve a very high percentage of the most popular videos.

They can still afford to serve the occasional obscure video from the origin servers.

[1] https://www.qqtube.com/blog/how-much-storage-does-youtube-ha...

[2] https://www.qnap.com/solution/petabyte-storage/en/