One issue about YouTube playlist is that the YouTube Data API has a daily quota limit. The default daily quota is 10,000 units per day. You can view the following page for more information: YouTube API Quota Usage.
If your website has a heavy traffic and it hits the daily quota limit, unless you are willing to pay Google, the YouTube API will stop working for the rest of the day, and the YouTube playlist gallery/carousel will not show up.
To fix the problem, in Wonder Grid Gallery, Wonder Gallery and Wonder Carousel, when adding a YouTube playlist, in the Add YouTube Playlist dialog, check the option Do not create a dynamic YouTube playlist gallery, instead, retrieve videos in the playlist and add them to the gallery.
By checking this option, the plugin will load the YouTube playlist and add all videos to the gallery/carousel at the design time. But of course, if you add new videos to your playlist in YouTube, you will have to go back to the Gallery or Carousel plugin and re-add the playlist.