Why would you need to play a YouTube video on the other site?
You can play their videos directly to your phone through either the YouTube app or through your browser app. No other site or app needed.
Unless, of course, you are wanting to listen without seeing/showing the video itself. However, as far as I know, it isn't possible to run them through Spotify. I searched the Spotify suggestions site and the info I found on it said they have no plans to allow/implement such an option.
That said, if all you are wanting is the stuff to play while your screen is dark/or on another screen, try the following, it should work for you...
1. Download and install the "Firefox" browser app from Google Play Store.
2. Once you have it installed, go to "YouTube" through the "Firefox" app and load your favorite videos (best to log in for that). Then just minimize the "Firefox" app. The music will still play, you will just not see the videos.