Is it possible to do a factory reset to rule out any software issues?
One technique I have used in tricky power on issues where to use a lab power supply and set that to the different voltages that exists in the device, lets say 5v that usually exists in all older electronic devices, and have the current limit set to 1A and see if it draws excessive current. And then try to power on the device while the lab power supply are helping out with extra current.
Then move on to next power rail and test. If the problematic power are found then let the power supply be on for a while to let a faulty component heat up so it can be felt by the touch of a finger, or it starts to tell you its location by smoke signals.
Many older devices use a tantalium capacitor at the reset pin of the CPU, to have it start after the voltages have been stabilised, and that capacitor have leaked and the voltage never rised enough to let the CPU start and a watch dog circuit where constantly restarting the device.
/Ubbe