Njaaah. The dirty bastids put the SMT module on top of the power vias.
*shakes the proverbial fist at the proverial whatever*
Second picture, top right chip: Find a datasheet or post a number/more clear picture for better advice.
Apples for oranges, though, I'm going to bet that you're either spot on or next to it about that being a charge regulator operating off of a 5V supply.
This 5V may also be able to power the device without the balancing properties of a luggy capacitance, such as is a Lithium Polymer cell.
When trying to rebuild it, be very aware that often in simple or small devices the empty-battery maximum charge current is left to the power supply, so start carefully.
Once it becomes a "seriously nobody knows" a first step would be conservatively set a lab supply, say 50 or 100mA at either 3V or 5V, depending on what you're "emulating".
Applying a short burst (< 5s) of maximum 250mW to something not expecting that voltage has a chance of not causing permanent damage, where longer exposure or higher currents often make things, chips especially, go *pop* quite efficiently and permanently. Fun as it may be, you might want to save that little giggle for when you want rid of it.
Important disclaimer: I do say "has a chance of not causing damage". Because it also still has a chance, though maybe not high, to go *poof* anyway and I am very unwilling to accept any blame in such cases.
I specify 3V for Vbat, in case you are not sure about the battery used, 3V covers most common set-ups post 2000. It falls within the operational voltage of LiIon/LiPoly (2.7V to 4.2V), LithiumPrime CR2032 types (2.2V to 3.2V), LiFePO4 (2.2V to 3.6V) and double alkaline (1.8V to 3.4V).