Also, this is basically their marquee feature. If they didn't get it right, shit goes wrong, real fast. Are we really saying they screwed it up? Gaming cap cards provide this exact feature with a lag-free experience, otherwise FGC people wouldn't fucking touch it. And you can do whatever the hell you want to the feed with those in Wirecast and such.
Plugging a DVR into the Xbox One is a marquee feature. You don't really care about lag on a DVR. They can delay the audio to match the video. You don't count on millisecond accuracy between your remote control and your DVR.
Plugging a game console into the HDMI input port is not a marquee feature.