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

[2]: https://yewtu.be/embed/xzTH_ZqaFKI

[3]: https://github.com/libredirect/browser_extension