No, the games would not have an "export video feature". The software included with the video tuner-enabled video card/ video capture card would have the ability to record the incoming stream to your hard drive, similar to what a TiVo does. Hook up the game system to the tuner card and fire up the software, telling it to record the feed coming into the card.
Overall the easiest option would be to record to a camcorder or use a DVD player with burning capabilities. You could then import the video via firewire with the camcorder or simply copy the video file from the DVD.