The pink slip is just a fast-track pass for immigration, normally given to premium pax.
Flew with AA JFK-LHR in the winter, didn't even say anything to the Purser (I guess he read the PIL), but after boarding he came down to me in eco, apologized for there not being room in biz or first, handed me an amenity kit and the fast-track pass (which I didn't need anyway) and wished me a nice flight. So yes, travelling staff are treated a "little" differently on US carriers.