I think USB data rate is too high to stay in-sync.
There is no quick solution for this. You could wait until LE is efficient enough, or you could buy a HiFiBerry DAC. Because HiFiBerry DAC uses GPIO instead of USB, it can handle high audio data rates more efficient.
As I learned recently, the Kodi process starts after autostart.sh execution, so this approach should work for you.
I just received some feedback. Bluetooth settings need an overhaul. So you will not get a fix for this very soon - sorry.
I'm not sure whether this issue has been fixed (new hardware). I've been informed the dev - stay tuned.
LE installs the right firmware at installation process. It's possible to update the firmware after this (using Raspbian is the easiest way), but as mentioned above, don't do this without a good reason.
Make sure you are using a blue USB port and try one of those test builds.
It doesn't make sense to add questions to a solved thread. All users with open questions should open a new thread.
The new Wiki is work-in-progress. Please continue to give us hints, which topics are missing. Thanks! 
This hardware is quite unusual. Most gamers use USB controllers. If you post the links to the scripts, we can hopefully give some hints.
You could activate log level 2 at advancedsettings.xml and read kodi.log after boot.
Better forget LE on that system. Install a regular Linux and autostart Kodi on it.
Chances are that the Python GPIO lib is missing. On LE you have to explicitly install that lib. It's contained at the "Raspberry Pi Tools" add-on. After add-on installation you can use the lib that way on your Python script:
import sys
sys.path.append('/storage/.kodi/addons/virtual.rpi-tools/lib')
import RPi.GPIO as GPIOTry Etcher to install on microSD.
To the Da Flex recommendation, using the 3.0 USB port is not 100%, but significantly increased the "success" rate, mostly on "cold" start
I think that's a proof of my theory, so data rate is the problem, not a wrongly loaded audio driver.
My suggestion: Take another microSD and try this audio-optimized LE version. Chances are that it can handle your high audio data rate better than the stable LE.
witlen English language please. Otherwise I'll start to delete your next posts.
Good choice to try an USB hub. This should provide enough wattage, and it's a universal device. Good luck!
Then boot a "Live" Ubuntu on your NUC. If you get the same issue on Ubuntu, go for a new tuner stick.
