In order to be effective against the numerous tracking techniques that are in use, the extension MUST block requests. If you aren't breaking many popular websites as a result of blocking their third party requests then it is your own privacy that gets broken. So this concerns me:

"In some cases a third-party domain provides some important aspect of a page's functionality, such as embedded maps, images, or fonts. In those cases, Privacy Badger will allow connections to the third party but will screen out its tracking cookies."