Why cannot I see any browsers in system preferences?
It looks like you may need to rebuild the Launch Services database, which is what the Mac uses for its "what apps can open this type of file" structure.

You can either do it in Terminal, using

/System/Library/Frameworks/CoreServices.framework/Frameworks/LaunchServices.framework/Support/lsregister -kill -r -domain local -domain system -domain user ; killall Dock


then restart,

Or Onyx (freeware) can do it for you - Maintenance tab > Rebuilding - check just the first box &amp; execute.