If you can render the paths and overlay them meaningfully, then the heatmap approach still works, but I'm not sure how easy it will be to accomplish in typical graphics software. The process I am about to detail will work in any linear environment where you can use floating point colors and perform meaningful math on them.

Render each of your paths as a white line on black. Then put all of the lines in a layer stack with a simple add blend mode. This will push your color way above white, so now you need to divide the result by the number of layers in your stack. That will normalize the image to put your brightest colors back to no brighter than white. Once you have that, you can use the technique of your choice to make a gradient out of it, adjust the gamma for a pleasing look, or whatever.

Now, about that bit in the first paragraph: A linear, floating point environment is difficult to set up in the Gimp or Photoshop. I'm sure it can be done, but if I were performing this job, I'd probably use Blackmagic Fusion, which was conveniently just released for free. Fusion is a node-based image compositor used to make feature film visual effects. As such, it's ridiculously powerful, and a little difficult to wrap your brain around at first. If you'd like to go for it, though, I can set up a template for you and walk you through how to use it to achieve your goal. You can download the software here: https://www.blackmagicdesign.com/products/fusion

And yes, it really is free.