I think the idea of a playlist is that you've already selected the songs and now they just play
In any case, this worked for me: go to the playlist, select a song, it goes to Album view, tap on the back arrow in the top left and the playlist view stays put.
Note that playlist view is different then track list view but they worked the same for me.
Also note that I kept the iPhone in portrait orientation -- if you use landscape orientation, you can tap on the cover art and get the track listing but I don't think you can get a playlist view this way.--
♬ Dragon of good fortune struggles with the trickster Fox ♬