A fun look at the meaning of "schleifen" and what dragging has to do with sharp. Also: a few surprise guests like "schleichen" and "schlecht"
schleifen, die Schleife, der Schleim, schleichen, die Schleichwerbung, schlichten, schlecht
The core idea of this root was:
gliding, slipping out
It’s very similar to the root *(s)lei- which is the origin of words like slide and slick, but the sources do not mention a connection.
In English, this root here is the origin of sleeve, sloppy and the family around lubricant.
The German side of the family is a bit bigger and got more creative with the idea. The verb schlüpfen was originally about gliding out (or in) an opening. From that evolved the nouns Schleife and Schlaufe which now are about a loop shape in a chord, but originally referred to the “sliding into” you can do with a loop.
The origin of schlau (cunning, smart) was the idea of being sneaky, making your way in or out somewhere.