Is my battery in my iPhone faulty, or something more?

My experience is you can triage it with these steps:

• Have you tried to do the force restart command while it is plugged into ACC even if your phone has no power?
• Have you tried using a different charger or socket?
• Have you noticed any pattern in what the charge remaining is when the shut downs happen? (If they are spread equally - less likely the battery, if they cluster around certain voltage levels - likely the battery)

Also, my tactics would be to wait about half an hour after plugging your phone in just to see if anything happens. My old iPhone 5 was useless for half an hour after I plugged it in when it was completely flat since it took a very long time to get past the weak area with lower voltage but worked OK at the top end.

Your symptoms could be logic board, but are much more likely a battery causing it instead of a short or electrical failure.