>In previous versions of OS X, if I went to ~/Library/Quicktime, I could see all the codecs used by QuickTime.
QuickTime codecs have never been stored in the username/Library location. As they are not user specific (or at least, they should not be). They are in the main Macintosh HD/Library and Macintosh HD/System/Library.
> For some reason, under Mountain Lion, this folder doesn't exist.
Lion hides the User Library directory by default.
Temporary access: Make Finder the active app. Hold the option key and click on Go in the menu bar. Select the Library directory.
Permanent access: Launch Terminal and enter the following:
chflags nohidden ~/Library/
Press Return.