I’ve had a YouTube playlist (with some other content) showing on a HA dashboard using the webpage (iframe/embed) card in my home office for a couple of years now.
Recently it broke due to a change Google made requiring additional headers for the referrer policy.
So, to get around that and also to make it easier to choose various options for how you want the video/playlist to work, I vibe coded a card which I now use.
You can get it from the HACS default repository, or check it out here: loryanstrant/ha-youtubevideocard: A dashboard card for Home Assistant that allows the embedding of YouTube videos and playlists