Good analysis! Since you charged the battery completely and tried it in the camera, and the camera did not turn on it is likely that something else is wrong with the camera.
Question: is the battery you have a genuine gopro battery? It is more likely for a genuine camera battery to communicate properly with the camera than a knockoff. I think the gopro 1st gen just straight up used 3.7v phone batteries to decrease the cost of production of the camera. If your battery is registering around 3.7v - 4v fully charged than the battery is probably working fine.
I would give the camera a look inside to see if anything else is broken. First look for any lose cables or broken components. See if there is any mechanical problem causing the electrical problem. See if the circuit board looks damaged (damaged circuit components). Try comparing your board to photos of others online.
Here's a partial teardown of the original gopro:
https://www.fictiv.com/blog/posts/gopro-evolution-a-teardown-through-the-ages-part-1
Good luck!