It a deficiency of web-player for a while. We couldn't do it on Android until recently, I think it took them about 6 years to implement it.
However it appears to have been implemented now and I was able to drag the song.
But I couldn't get the playlist to "scroll" as I dragged it, so I had to drop it where it was still in view, then scroll down a bit and repeat until it finally got where I wanted.
Being able to "sub" a wrong track in a playlist is certainly a feature I would like to see, and for this to keep the play-queue exactly as it is other than the track you sub. So often "it" gives you a remix or live version you didn't want.
("It" being the playlist builder, which isn't actually Spotify itself).
It might be worth trying to log out and back in the web player. Also if you're not seeing a change you make in your playlist immediately, try refreshing the page. Playlist alterations like moving/deleting songs should become visible then.
This is due to the nature of the web player.
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