Private playlists have a variety of uses, from keeping your personal and sometimes embarrassing songs to yourself ... to your top selections for upcoming artists you have your eye on.
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Private playlists remain private, until you share the link. Once this has happened anyone with the link has access, which means also anyone can get the data from Spotify's API. There’s little we can do about it this as the data comes from Spotify.
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To help remove old unused (and private) playlists from Chartmetric we implemented a code to delete playlists from the database, when we detect that the playlist is not accessible any longer.

If you still have concerns you could make a new private playlist, move your songs there and delete the songs in the other playlist, followed by deleting the old playlist. Once you have done this be sure not to share the link and your playlist should stay private.