iPhone 4s won't turn on without charger connected

Helping a friend with his iPhone 4s that he wants to revive. I've already suggested an upgrade, but he insists on keeping this old dog.

Symptoms:

Phone won't power on with battery alone. It will power on and almost complete a boot while connected to a charge cable. However, it will power off just before connecting to cellular service.

My friend says it's never been wet, and indicators all look white still. He said he thought he smelled smoke coming from the phone just before it died.

Step I have done:

I have replaced the battery and the charge cable. Neither fixed the issue.

I have visually inspected the PCB components looking for a blown IC somewhere, but nothing jumps out.

I read here that you should check for continuity on the first and fourth pin of the battery connector. I do show continuity between those two pins, which is supposedly indicative of a short somewhere.

Looking for experienced techs to weigh in here and confirm that I'm likely over my head on this particular repair.