I'm not sure what you mean by a system file. Tbh, going through the registry is probably a little extreme. I would first try a proper uninstall through Control Panel/Programs and Features. After you do that go to these 2 locations and delete the Firefox folders
C:\Users\Dad\AppData\Roaming\Mozilla
C:\Users\Dad\AppData\Local\Mozilla
You'll have to enable hidden files and folders to see these, and of course you'll substitute your directory names in the appropriate places. After you do that, try reinstalling Firefox and see if it works. If it still doesn't work, do all of the above again, then search the registry for all examples of mozilla, and firefox then delete them(be careful if you have other Mozilla products on your system), then reinstall. If that doesn't work, I'm out of ideas. Maybe do a virus scan in case a virus is trying to limit your browser selection for some reason. I've never heard of anything like that before though.