The only disadvantage of display port that I can think of is that it's harder to find capture devices (or at least Linux compatible ones) that have display port. Generally no one cares about that stuff though. I use capture cards because I sometimes do stuff involving other computers and it's 1000 times more convenient to have a vlc window floating around with the display output.

No one seems to have this use case besides me, I'm just glad that there exist capture cards fast enough to do playback in real-time and not a 5 second delay. I don't stream.