I had a look at the original question and I have a couple quick follow ups just to be sure we're not running down an issue that could just be the battery.
Did you ever try a known good battery with this board? Could just be me trying to get the overall gist of this, but I am not apt to jump to a board problem, until I know for certain that it's not a replaceable component.
The pictures you posted in your update are what we would call "No Stuff." All that means is that the board has the place for a component to go. The schematic may even specify something that goes there. But the final board design does not have any component in that location. So those spots being empty is totally normal. You can usually tell because the pads are too shiny and perfect.
As for charging on this board. It's a bit complicated. Charging itself is handled by the ISL9240HI (yellow in the pic). But the TI CD3215s (blue in the pic) previously mentioned also play an integral role in negotiating the necessary 20 volts from the charger. But if you can run it off of wall power, with no battery. Then you're getting the correct voltage. So I would be more likely to think the ISL is your issue if it's not the battery itself.