I'm guessing that you've already taken a look at Apple logo then blue screen since you mentioned that you checked for long screw damage and most of that thread talks about screw issues.
Since this seems to have begun occuring without any obvious precursor (droping the phone, water damage, repair gone wrong) it could be a software issue. I read through this article and it seemed helpful: http://vkrepair.com/iphone-5s-blue-scree... There is a section on software solutions that recommends the following.
- Press the home button and the lock button at the same time and hold them until your iPhone 5S shut down.
- Power on the iPhone.
- When you see the Apple Logo, press the volume up button until it boots up. This might take a few tries. You may have to hold onto the volume button for a few minutes.
If you're still having trouble, you may have to reinstall the iOS on the device completely. To do this;
- Plug the iPhone into your computer.
- Power it down by holding the home and lock buttons.
- When you see the Apple logo come up, let go of the lock button but keep holding the home button.
- The iTunes logo should come up on the iPhone.
- Restore the iOS with a new set up, do not use a backup.