I don't think that this is related to Xbox Live.
Yes. I will make some test to see if I keep the screen open, if the problem will occur.
The screen is turning off, but if I click the home button, the pause menu of Minecraft is there.
The issue is now resolved. The server has started.
If you need the server owner username, I can give it to you.
Now for 3 hours.
I can confirm the same bug here on Windows 10