First I would start with a new battery. While you have it open check for any water damage. Next I would have the logic board tested. Definitely back up everything if you can
What about the connectors on the logic. If you get something to magnify them so you can see better do they look okay. It can get very frustrating some times but what i would do is buy another one from a online app like (Craigs list) or (offer up) that either is locked or has a broken screen so it's cheap but you know it still boots fine. Just make sure you check the IMEI online or call the carrier with the info so you know it's not stolen. Then I would start swapping parts one by one.
I have a feeling though that if you have swapped batteries multiple times and tried to restore it that your have some kind of a software defect or something with that logic board has been damaged.
What was going on with it when this first stared. Did it just happen one day or were you updating it and the battery died maybe?