First, there are two ways to embed YouTube videos in your Blogger blog posts :
1- Either through the video icon you have in the Blogger word editor as the screenshot shows, but the problem with this is that it does not always work.
2- The second way to embed YouTube videos and which always works is to grab the embed code of the video and paste it in the HTML editor of your post. Don't freak out , it is very easy and here is how to do it.
Head over to the YouTube video you want to embed and look for the button " share" under it. Click on it, then click on " embed" then grab the highlighted code of that video.
Now back to your blog word editor. click on HTML button and paste the code where you want it to appear in the post.
I hope that helps. If you still have any problem with it, please let me know and I will make a video on how to do it.