I am still having this problem. I am using the iframe HTML code that YouTube supplies:
<iframe width="560" height="315" src="https://www.youtube.com/embed/videoseries?list=PLzlNKjoB23LmDd78q8rtSYU1TeLD2FFuM" frameborder="0" allow="autoplay; encrypted-media" allowfullscreen></iframe>
The code works fine if I put in the code for a specific video, but if I make it a playlist, I get the Video Unavailable error. I have also tried using the shortcode (including the shortcode you provided above), but to no effect.
I have noticed that I am suddenly getting the following error in wp-admin:
WP YouTube Lyte got the following error back from the YouTube API: accessNotConfigured (Wed, 01 Jan 2020 20:23:45 +0000)..
I haven’t made any changes to the API access codes so I am not sure why this has just cropped up, either.