It definitely IS possible, the internet says that Autocad has a tool called “Overkill” that sounds like it does exactly what is needed to fix this file, but I couldn’t find any other software with similar functionality.
SolidWorks almost has it, with its “Repair Sketch” command, but it doesn’t seem to have any options (surely it is utilizing variables, Dassault apparently choosing not to let the user edit them is really annoying). From my quick look at it, it basically automagically removes overlapping and almost overlapping lines, or it doesn’t. It also seems to give very little indication of what it’s actually done, if anything, after you execute the command.
Rhino has a tool called “SelDup”, which works perfectly (IMO) for finding lines that are exact duplicates of each other, but doesn’t seem to have any provisions for finding lines that are very close to each other but don’t quite overlap. It also doesn’t find lines that do overlap but don’t have the exact same end points - it only finds exact duplicates.
I was able to clean that file up using Inkscape to color the states (this might not have been necessary) and export a fairly high resolution PNG (I guess Rhino could have been used to do this step as well), then use Vector Magic to trace the PNG and create an SVG, then use Rhino to delete all the overlapping lines. That’s a lot of steps and software, and the vector to raster to vector conversion obviously degraded the detail.