I open dev tools
I look for errors in the console
I inspect the source of the blog post to see what wordpress did/actually generate.
Theres too many what ifs to give a straight answer.
Depending on which version of Wordpress you are running, what theme, did wordpress strip the HTML content out, are you using a shortcode for the embed an the shortcode is not working properly.
This is more a Wordpress problem (how to use iframes and script in posts/pages) than a Twitch problem.
I integrate it with the theme at a different level, writing my own plugin or theme functions and use a wordpress shortcode.