If it still happens after a clean boot, that makes it much less likely that a background program or service could be causing the problem directly, or interfering with the game closing. In that case, though, I'm not sure why there would be a time dependence...
Can you restart the game directly from the executable when Steam says it is running?
If you reboot, exit out of Steam, start the game from the executable and play more than 30 minutes, exit and after 5 minutes restart Steam, does it list the game as running?
Especially if the problem only happens if the game is started through Steam, you could try to
Refresh Your Steam Files and reboot your computer, to see if that fixes it. With the Task Manage not listing the game running but Steam doing so, it could be a problem just with Steam.
