In the case of iA Writer, they licensed a commercial font called Nitti Light for use in their application and their application only. That's why you cannot copy it into FontBook and use it elsewhere. Try Cousine from Google Fonts for a free font that is quite similar.
OS X normally will not display a character in the Private Use Area unless you manually set the font in the app. I understand this is intentional, since every font could have a different glyph at the same PUA code point, and it only makes sense to display something when the user specifies the font.
I think it should also work when the right font is selected via CSS or other markup in a document being read.
Some earlier versions of OS X did not have this "feature" and did display PUA stuff. I don't know how they chose which font to use when more than one had something at the code point selected.
Best Answer
Open Font Book.app and choose File → Restore Standard Fonts….