You could not test it before the new screen and even a new battery made no difference. If the Date/Time suggestion above does not work, it sounds like whatever caused the screen damage also damaged the logic board. You could have caused some Pry Damage or damaged an FPC connector during your repair.
Could you somehow fully charge a battery (in another 5S or SE) and then install it in the device? That would help you understand if the problem is charging related or elsewhere. Charging problems are usually related to a Tristar IC being defective or PRY / FPC damage mentioned above.
This is a difficult repair and unless you have the proper equipment or experience, I would recommend that you find a repair shop that can do micro-soldering repairs.