The DVR box has no real concept of latency (you pause and 1000ms later it pauses and you would not care, 100ms is fatal for most games that require any kind of reaction) or need for image processing power though (give or take simplistic things like scaling and basic filters it might do) so it can afford to do things like that. Games which require real time input/output of controls and the corresponding graphics which it renders itself have a harder time of things. Or to spin it another way if someone came and said I have two monitors but if I disable one the fps in my game goes up would you say "well durr"? Same difference really.
The console makers probably could output multiple outputs with some simple extra hardware or hardware geared from the ground up for it. Unlike camera makers not outputting real time HDMI though (my cheapo Chinese gopro does it fine, my not quite professional camera does not, yet it has HDMI out) I am not calling conspiracy -- it is extra work, extra hardware and potentially more trouble down the line for someone a vanishingly small number of people care about, and even if more did there are still splitters and converters that cost next to nothing. Similarly the DVR box probably has a use case for it as people do want to pipe it to their bedroom/kitchen/second room/garage/shed TV, I know the cable/satellite companies want me to buy a second box and pay more for it but DVR box makers are less concerned.
Edit forgot to answer the question on adapters.
If you buy the cheapest piece of junk the quality might not be great, have a look around to see what goes as I am not really up on that world right now (I try never to go backwards in video quality and screens cost nothing when you can fix them).
I don't know what goes with wii u games and the outputs. On the 360 various games did render differently (text that is visible on a large 1080p screen might not be OK on a small NTSC interlaced affair, the first Dead Rising being the easiest case to look for) though so that might be a problem for some things.
Equally I don't know what DAC, or indeed if the graphics chip inside the wii u does it itself, the Wii U uses to generate the RCA output. If your converter is better (certainly possible) it might do better -- plenty of consoles have lousy audio DACs and can be bypassed/modified and I doubt Nintendo was overly concerned with this.