If this is too off-topic please just ignore it (and don't DV).
I want to embed YT videos on a website, but we would like them to be as private as possible. For example, DDG has a player which shields the watcher from YT's data hoovering.
Any suggestions? Perhaps the search terms I'm using are wrong?
You can embed videos from an Invidious instance instead; on a video's page[1] there's an "embed video" link[2] you can use. The instance can be one hosted by you if you don't trust public ones, and you probably want to enable proxying by default if you don't want your clients to stream the video directly from Google's servers. You can also use a browser extension like libredirect[3] to automatically replace YouTube embeds with Invidious ones while browsing the web.
[1]: https://yewtu.be/watch?v=xzTH_ZqaFKI