I’m trying to rebuild the original curve that generated a mesh.
How can I create points that are exactly in the middle of these rings and then connect them together to be a curve?
Actually two problems.
The first one is easily solved. You have to set the “Attribute Statistic” from Point to Spline.
The second one is a bit worse. The problem is that your curve indexes seem to be out of order.
When looking from front view: For whatever reason the index of the segment left of the center segment appears to be mixed up with the index of the segment in the center.
Perhaps you can get away with simply deleting the offending segment. Or fix it by moving the segments around by hand.