Most of the logic board is covered with shields anyway so unless you removed them, it will be difficult to find anything wrong visually, especially in relation to charging.
The other logic board works so that clears everything else with the phone, including the battery. Clearly the problem is the logic board. You need to verify the main power rails to determine what is at fault. Has anything else happened to the phone other than the shattered screen. That impact alone could have caused some issues.
We don't have much info to go on so this is pretty open-ended troubleshooting. Unless you have access to schematics and troubleshooting tools, I recommend you seek out a qualified shop that can do micro-soldering repairs. The iPhone 5S is still a worthwhile phone to repair, no need to put it in the landfill.