Hello,
The problem might be that you are not using the YouTube Component.
It’s located in the Components Pane drop down on the right side of the GWD window.
I’ve attached links for 2 good resources below.
The YouTube Component:
https://support.google.com/webdesigner/answer/3232992?hl=en
Here is a good example of adding YouTube content to your website using the YouTube Component:
Banner Ad with YouTube Content
https://support.google.com/webdesigner/answer/3265056?hl=en
Hope this helps!
- CN