How to hook to the end of animated curve bevel

Hi, I’m trying to create an animation where a thick strip of paper is wound on a tube.

What I did is I made a bezier curve spiral around the tube, gave it a bevel in the cross section of a strip and animated the bevel end. Then I animated the rotation as well to get the effect that the paper strip is being wound on the tube.
I attached a static piece of paper to the end of the bevel and animated the x position along the tube.
Probably sounds a bit confusing, but I’ll all make sense in the blend file here:
strip_on_tube.blend (832.8 KB)
Link to google drive folder with blend file and a few videos showcasing the problem.

It’s kind of fake but it gets the job done and I couldn’t figure out a better way how to do it.

As long as the path is a perfect spiral it all works well. However if the path is different the animation needs to be keyframed at almost every frame to get a smooth animation. Since I’m looking at doing a lot of these animations and I expect some revisions I’d prefer a better more flexible solution.

My idea was to hook the end of the straight strip of paper to the end of the curve but this seems impossible.
The only alternative I can think of would be animation nodes or python, however I wouldn’t even know where to start there.

I’d be glad if anyone could point me in the right direction on how I could best solve this.