I've been having this same problem for the past 3 years.
Spotify Connect used to work fine with large playlists that had several thousand songs. Then all of a sudden it would only play the latest 200 songs that were added to the large playlist over and over again even with shuffle. Then about a year ago, now when you try to play the playlist over connect, nothing happens. When I have music playing in the desktop app and I try to play a large playlist with several thousand songs using the mobile app and I press on anything in the large playlist, nothing happens. The only way I can get the playlist to play in the desktop app is to switch the Connect playback to my phone, play the playlist, and then switch the Connect playback back to the desktop app. But it won't play everything from the large playlist, it's a very limited work around for playing large playlists using Connect.
It doesn't just happen on desktop app, it happens too when I want to use mobile app to play music on Chromecast or an internet connected speaker. I wonder if there is some new technical limitation and that is why they have made it so you can't use large playlists over connect anymore.
So pretty much I can never use the Connect feature unless I just want to listen to a certain album. My main playlists are too big with several thousand songs. The whole point of Spotify Connect is to be able to use it as a remote from when you are away from your computer and want to control it from your phone or you want to play Spotify on Chromecast, or an internet connected speaker using your phone.
If I want to listen to a large playlist I have to physically use the desktop or mobile app and listen to it there. I can't use Connect to play it. And connect doesn't even give me an error message. You press on the large playlist and nothing happens over connect. How are regular users supposed to know what the problem is? I hope this is fixed in the future.
MattSudaSpotify Star
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