Not necessarily. Youtube makes extensive use of third-party CDNs. A lot of the videos aren't coming from their servers at all. I believe that's also why it's so hard for them to embed the ad directly in the video. They instead having to rely on splicing the ads client-side, which makes it possible to block.

Disclaimer: I work at Google but not at Youtube and have no idea how things work really. This is just based on some info I read online.

Yeah they give caching boxes to ISPs as far as I can tell, and videos are served from there if they exist in that cache. About 8-10 years ago, they had an issue with that and they'd serve you the wrong video because your neighbor had watched something and it was in the cache. Literally title of the video wouldn't match what is playing.