Experiment with Geometry Nodes. Quite tricky. Maybe somebody else would like to try it. Without Geometry Nodes I think it is possible, but worse. (Note: the rope length is not constant - an exercise for the student.)
P.S. I really wanted a ribbon instead of a rope, but I couldn't control the twist.
Fourth approach, back to no Geometry Nodes. Modifiers and tricky animation. Looks a little better - close to what I wanted.
Thanks 
spikeyxxx . My video "04" above (approach #4, and my approach #2) are very similar to what you describe. I just slid a long Plane (ribbon) along a curve, to get both the straight tail and spiral part. With animation I translated and rotated everything to get the right appearance. (I like using transparent shaders but didn't need to in this case.)
A single plane helped a lot to get a smooth transition. I struggled with the spiral Curve until I found the setting "Twist Method Z-Up" (it's a mess to do this by hand). The animations are a little tricky because some of them work against others.
I finally got more or less what I was going for. The final result can be seen in
