It ain't really an embed more like a link to itch.io. Is it possible to really embed the game so that is is playable into my website?
Viewing post in how to use the widget/embed for your game
I actually need the same thing. I was thinking to use put it into an iframe, then scroll the page by javascript, so only the canvas can be seen, that has the same size like the iframe on my webpage. However, I didn't try it yet, and maybe itchio uses a "break out of parent iframes" code that would prevent it.
If you have a website, then you could also host the game yourself, on your webpage, or register any ad-free free webspace and host it there. Download speed may be an issue on many free webspace hosts however.
Either way, itch.io would do well in offering seamless integration with nothing more than a discreet "Game Hosting kindly provided by itch.io" link.