Way to go Philly_Phan! He's right. There is a difference between a recovery mode restore and a DFU mode restore. In the absence of 5s long screw damage from a recent screen replacement (mixing up the screws) or failure of the tristar/U2 charging logic ic, or battery connector area filters, the most likely cause of 5s BSOD is simple software corruption.
A DFU restore will resolve if there is no underlying hardware defect. Use Philly's instructions to make sure you are initiating the restore from DFU (blank screen) and not recovery mode (connect to iTunes logo).
If the restore still fails, then the most likely candidate becomes damage to the tristar (U2) chip on the board from use of non-Apple charging cables. With U2 failure, you will typically see iTunes error 9 (dropped usb connection which is one of the functions of the chip) but other errors are possible. The U2 chip can be changed on the logic board by a qualified microsolderer experienced in iPhone board repair.
Please note that pursuing independent repair options may void your Out of Warranty Swap option to turn in your phone and pay $269 for a refurbished iPhone 5s, although I have never seen that happen. The common value of a good condition 5s currently is less than $269