Problem: You open your music app on your iPhone as usual. You try to watch a video in landscape mode. You can't. WTF.
Solution: Use the video player. See below. Seriously, that's the only way to do it. Unbelievable.
Details on how to use the video player:
1) Open this thing:
2) Something like this will come up. Click "Music videos" in the lower right.
3) Click on the video you want to watch
If you have a better solution, maybe one that lets you use playlists, please share it in the comments. For the love of all that's holy, please share it in the comments. Because this workaround sucks.