In that situation I’d favor self-hosting - it’s easy, friendlier for users, and unless you’re getting a huge amount of traffic not very expensive. Swimming upstream against YouTube’s business model is just locking in time spent worrying about what you missed.

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

Peertube lets you host them yourself.

Hi, you can embed it using simple HTML! For example, you can try this: <iframe src="https://piped.video/embed/jNQXAC9IVRw" />