I'm a big Spotify fan. Been using it for a couple of years. But it took me a while to figure out all the nuances of how to navigate the different options in the app. (The PC is easier to navigate.)

FWIW, I am able to rearrange the song playlist on both my iPhone and my PC, so I'm wondering if the option is there, but hidden. On my iPhone, when I open a playlist, there are three horizontal dots in the upper right corner of the screen. I click on the dots and a menu appears to edit, rename, etc. the playlist. If I click on edit, I can delete songs or rearrange them by clicking on the the horizontal lines o the right of each song and drag them up or down to rearrange.

You then click Done in the top right corner to save the changes.

No idea of this is the way it works on Android. Hope you can get it to work.