Here is a result I made. 3 or 4 lines are missing, but it could have been dealt with. A little more than just 1 click though.



Since you want to keep the breakpoints of the polylines, that is what kindof makes the offsetting hard.

So I joined and rebuilt the original file (Rebuild to remove all the corner points)

Then I extruded the new curves and shelled them on centerline. I could then Boolean together the shelled solids into a single solid.
(This is where I lost 4 lines. they would need to be addressed at the extrusion level to get good solids to bool)

From the single solid, I can extract the bottom edges to create the thickened curves.

I can now bring in the original file and extract the points I wanted to keep (As waypoints I guess?)

Anyway, I can make a video if this file looks like what you are after and you wanted to see more.