What you seek is commonly called a centerline trace. My research shows me that nothing exists for Adobe Illustrator and there may be an Inkscape extension or feature which works, but I did not pursue that thread.
I found a video from a creator who uses Silhouette Studio (craft cutter) presenting a similar requirement. The solution provided in the video is an online resource called Rapid Resizer.
It appears that one must begin with a bitmap and select either outline, which you have and do not want, or centerline. It's important to note that centerline trace will not work with T-intersections, but this online resource appears to resolve that by terminating the path short of the intersection. This is apparent in the linked video.
Above image is a screen capture from the linked video.