I'm not familiar with AMD's recorder tech, but if it's anything like the nVidia one then it's simply piping video output to a second output stream that encodes the video on the fly. This is the most efficient path as it avoids any overhead on the CPU or re-routing the same contact back to the video card that just rendered it for re-encoding.
If this is the case with AMD's stuff, then it's not picking up Flash because the video card isn't rendering the game. Flash leans on graphic cards for some acceleration, but the card doesn't directly render the game. In other words, the recorder doesn't "see" what's going on in the Flash window.