Historically asciinema was not easy to embed in React because you can't have multiple copies of React. So my team at the time wrote and open-sourced an embeddable alternative to it. Unfortunately that project seems to have disappeared by now.
But now that asciinema is no longer in React maybe it will be possible to embed now. See https://github.com/asciinema/asciinema-player/issues/72#issu....
Historically, you could embed an animated .gif into a web page going back before Y2K, and there are tools to make such a thing from recording terminal sessions. No Javascript, no third party websites.
FYI you're not forced to use asciinema.org for hosting the recordings, it's fully self-hosting friendly: https://github.com/asciinema/asciinema-player/#quick-start