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I loved the script, but it doesn't work with private windows in Firefox, only in normal windows. Am I doing something wrong?

To be more specific, the script shows the resources normally, what fails is only the download of the media

I'm not sure if what you described is what I fixed, please update and try again.
Regarding private windows not working in Firefox, maybe you can check if your browser allows extensions to work in private mode.

Yes, Tampermonkey is working normally with other scripts in Private windows. Does the download work normally in a private window on your machine? I tried looking at the browser console when trying to download in the Private window and this is what I got (this error does not appear when I download in a regular window):

Yes. In version 2.18.5, scripts can download resources for me in private windows in Firefox.