I think the easiest way is to use a oscilloscope to check if the max8724 at least does something. On PIC 150003-5 you mention that 1 component has 1.5V at a good board and 0 volt at a bad board. Could you check if this is a dual p fet. What is it's marking ?
If it is a dual p fet, then likely there is a square-wave present on those pins. Because 1.5 Volt sounds to me like a square-wave.
EDIT:
The SI4435 is a P-Mosfet. This means that the pins called "s" or "source "must be more positive then the pin called "gate" in order for the mosfet to conduct electricity from source to drain.
The SI4890 is a N-mosfet. This means that the pin called "gate" must be more positive then the pins called "s" or "source" in order for the mosfet to conduct electricity from drain to source.
Looking at picture 150003-1 : I have difficulties interpreting what you mean. If i read correctly , it means that on the defect board PQ53 does not switch when looking at how the copper traces connect to the row of capacitors. You must start from the ac pin and look from there on the mosfets that are connected to the ac adapter input.
When looking at the pictures, i notice there are 2 battery connections ?
Are there also 2 max8724 ic's ?
If not then the charger is switched between the 2 batteries.
If there are 2 battery connections, have you checked what the pcb does when connecting 1 battery to a connection at a time ?
Let me know, i am getting curious