Cloth is more for ... well cloth, but a good start, especially when a loop is wanted. In fact is is more like a flag waving in the wind with some turbulence wind from the side. Centrifugal powers version rotation and inertia. But once you have the flag waving, rotate it. The loop could be done with an Animation Clip.
I have attached a sketch based on your file. Sketch means here, that it certainly need the “time” to be animated smoothly.
You will find a Null object that rotates the whole thing. This could be more animated already, following the hand of a “towel mover”, here it is simp lied.
The other null named with a number is an summery of the four states that I have used here, it should show you always a 1.0 (100%), to have not weird towel stretches.
So animated the rotation first and the time for one loop (360*(loopframes+1)/loopframes) linear, then the states of the towel based on the sliders of the PoseMorph.
Again this is a sketch and needs the hand of an animator that likes to invest more time than I have for that example.
If all sliders are to zero, then you can comfortably texture the “towel”
I have tested a little bit and a wave deformer with some adjustments creates just a nice “distortion” to the over all movement. There are more options to move points… to get the detail moves.
All the best
Sorry file is too large I can’t compress it smaller.