I'd like to convert this shape into a path from the center so I can easily adjust the weight of the graphic with a stroke.
Drawing it anew won't preserve the round elements perfectly.
The techniques described in: Convert outlined stroke to single stroke or Technique for finding the centerline of a curved object? are about as good as you're going to get without manually redrawing the shapes yourself.
For anything with a consistent line width then Illustrator's Offset Path will generally give you a good result. You can see from my example that there are some issues where the paths meet, but you still have a good result:
But in this case there's no real need for any of that; the shape is made entirely from circles vertical lines.
This took me about 10 seconds to redraw:
The paths aren't completely accurate but if I'd have spent maybe a minute more measuring the line widths then it'd be pretty spot on. Make use of some guides at edges of all the lines like this and you can be pretty accurate with minimal effort:
Make a rectangle Select the rectangle and make Art brush Draw a Line and apply the brush strokes Draw a circle and apply the brush strokes Copy the circle and the line and reflect Drag the copy circle and line little to snap the circle Select the circle and delete the below node to make the arc shape Extend the line to align both same height