Could you please try this:
LibreELEC-Generic.x86_64-8.0-Milhouse-20160814210359-#0814-g17fb8db.tar
This are the most recent buils from one of our devs. Your issue might be a driver and modesetting error, which already might be fixed in the build I linked above. Those builds mostly go mainline, so there is a good chance if it's fixed there, it will be fixed in the next version.
Please test the current v7.90.004 alpha build. If this works, great. If it doesn't, at least you're using something that is still being actively developed. At this stage in the Jarvis lifecycle (all Kodi development stopped 5-6 months ago) we are unlikely to go back and fix anything, even if the issue is related to drivers in the OS (probably) and not actually about Kodi.
I have a Celeron NUC with LibreELEC 7.0.2 & have had NO problems.
Incidentally, I have LibreELEC on microsd card on a Beelink miniIII. again no problems.
If I plug in the nuc in my pc monitor display is ok.
[hr]
This are the most recent buils from one of our devs. Your issue might be a driver and modesetting error, which already might be fixed in the build I linked above. Those builds mostly go mainline, so there is a good chance if it's fixed there, it will be fixed in the next version.
I tried this build: LibreELEC-Generic.x86_64-8.0-Milhouse-20160815210333-#0815-g44a05fd.tar
Still no tv output 
could you please try again and probably provide a debuglog?
I guess this is a migration and not a fresh install, isn't it?
Could you try to do a factory reset while at LibreELEC? SSH in and do
mv /storage/.kodi /storage/kodi-bakAnd reboot after that.
All your settings and addons will be gone and everything is like after a fresh install. But no worries...everything is copied to the "kodi-bak" folder and restoring it is easy.
Sorry to see you are having issues Markus. Whilst I do have video on my devices the video playback is stuttering. Going back to OpenELEC v6.0.3 resolves it but seeing as your NUC uses HD 4400 it is obviously in the Ivybridge family of chipsets and that seems to be the cut off point for issues as you can see in my post here.
Can someone share output from "cat /var/log/Xorg.0.log | paste" and "/usr/lib/kodi/kodi-xrandr | paste" please. It has nothing to do with Kodi itself so debug logs from that aren't going to reveal anything. It's mostly likely to be related to the Intel driver in the Kernel.
xrandr --output HDMI1 --mode 0xe0I'm wondering if 
that results in output? (59.94) and if that works, try the "0x48" mode (60.00) again.
