Is there a plugin (for any browser) anyone can recommend? An app?
It seems like all browsers got rid of this easy resource tracking feature (Chrome too).
If you don't mind payware, I have Jaksta and it downloads just about any non-DRMed streaming videos. Simply go to the webpage with the video and it'll detect and download it from the beginning, you can close the browser tab once it's started and it'll continue to download., it'll even name the video after the page title.
You can see still browse resources in the web inspector.
To download a file, you can copy the URL from the right sidebar, paste it on the location bar, and press ⌥↩.
The web inspector can be shown by pressing ⌥⌘I if you've enabled showing the develop menu in the advanced preferences.