The PS3 cable included with the Elgato is to get around the fact that the PS3 doesn't allow recording over HDMI for some stupid reason (even the PS4 fixed that, though Sony dragged their feet on that for some time). It is nothing more than a component connection (and I suspect it works with the PS2 as well, though I've not tried this).
I've had zero issues with my Wii U and Elgato. It obviously doesn't work if I'm running Netflix, but that's because Netflix turns on the HDCP mode (which is otherwise disabled), but apart from that I've recorded videos just fine. I'm not using any splitters, however.