Sorry for the late reply, I was off for the weekend and could only test it yesterday. Thanks for all your answers, and even bigger thanks to you guys at Unbroken Software for lovely LauchBox and your devotion to it.
Indeed BigBox streaming works when I connect a keyboard to the Nvidia Shield TV: I bought a Logitech K400 and I was probably lucky that I had no other choice in the shop because BigBox also didn't respond to the cursor keys first, but it all started working once I clicked the left mouse button.
This makes me think BigBox simply loses focus when it starts. I repeated this several times and it turns out that even the gamepad works once I click the mouse. Maybe this is the only problem (and maybe you could look at it when you have the opportunity - on a PC you probably wouldn't even notice it), afterwards navigating works well and the emulators run fine with just very little additional, noticeable lag in games. Not very surprising because it's the same with the PC games I stream this way.
@AutumnSounds, you're right, all these streaming solutions are a sort of "full screen capture" and usually you shouldn't even have to bother about streaming at all in LaunchBox development, except such small issues like avoiding focus loss as it appears now.
What I will do within the next weeks is trying different streaming software (Moonlight, Remotr, KinoConsole) on all the hardware I can access or get from friends for testing (iPad, Amazon FireTV, several smartphones and some of these cheap, chinese Android boxes should be within reach) and let you know if this is a Nvidia-specific issue or a general one, then we might have some facts to discuss.
Just because this tread devoloped a bit into a general off-topic discussion let me add my two cents to make my intentions clear:
I always had a HTPC in my living room for far more than a decade (and spent a fortune on upgrades in all that time) but I got rid of it last year and I don't miss it. Usually it was either not powerful enough to do everything, or it was too noisy to have it in the living room. It was always a foul compromise, and I'm now pretty convinced that this is a thing of the past with the entire entertainment industry and technology changing so fast and so dramatically.
I really like the NUCs, but it's not true that you spend just a hundred bucks because then you may have the box, but you don't have any RAM, no mass storage, no remote control, no gamepad and no Windows license yet - and how do you think the Dreamcast/PS2 emulators or Dolphin will perform on a Baytrail-"Celeron"-Atom and it's integrated graphics?
The great thing about LaunchBox is that it's so flexible and can be used to emulate whatever the PC hardware can handle. For MAME, SNES and the other old staff a RaspberryPi would be good enough - and then again you would spend far too much for a solution based on an entry-level NUC. Ok, you would miss LaunchBox then...
I am now moving more and more functions to a "home server" (what a big name for a normal PC) and this has just recently started making more and more sense for me. Not everything is perfect yet, but the progress is amazing. These cheap "thin clients" are a blessing: with Kodi or Plex including Live-TV recording and media access on your NAS, streamed games, streamed music and video from Netflix/Spotify etc., and lots of native apps for whatever, there's no need for "more" and you don't even have to worry when your TV's "smart" functions are no longer up to date (usually on day three after the next year's model was released).
When there is enough technical progress after a few years, I will simply replace the cheap box with another one.